Quadratic Equations

A quadratic equation has the form $ax^2 + bx + c=0$, where $a, b$, and $c$ are real numbers, and $a \neq 0$. The $ax^2$ term is called the quadratic term, the $bx$ is the linear term, and $c$ is called the constant term. Consider the quadratic equation $x^2 – x -6=0$, if we substitute $x=3$ in this equation we find that the equation holds true because $3^2-3-6=9-9=0$. Similarly if we substitute $x=-2$ in this equation we again find that $ (-2)^2 – (-2) – 6 = 4+2-6=0$. These two values of $x$ are the only two numbers that satisfy this quadratic equation, and are called the roots of the quadratic equation. Sometimes the roots are also referred to as zeros of the quadratic equation, in other words the value of the quadratic expression is zero at these values.

Factoring Quadratics: Trial and Error

The roots of the quadratic equation $x^2 – x -6=0$ can be obtained by factoring the equation into a product of binomials. Let’s assume that the binomials are $x+r_1$ and $x+r_2$, our goal is to find the values of $r_1$ and $r_2$ such that: $$(x+r_1)(x+r_2) = x^2 – x -6$$ $$x^2 + (r_1+r_2)x + r_1r_2 = x^2 – x -6$$ The two expressions above are identical and therefore the coefficient of $x$ terms must be equal or $r_1+r_2=-1$. Similarly, the constant terms must also be equal, therefore $r_1 r_2=-6$. By using trial and error we find that the values of $r_1$ and $r_2$ are $2$ and $-3$. Therefore our quadratic equation can be rewritten as $$x^2 – x -6 = (x+2)(x-3) =0$$ For the above equation to hold true, either $x+2=0$ or $x-3=0$, which tells us that $x=-2$ and $x=3$ are the solutions to this quadratic equation.

Factoring Quadratics: General Case

When the coefficient of the $x^2$ term is not 1 we follow the steps described below to factor the quadratic equation. I will use the example of $2x^2-5x-33=0$.

• Multiply the coefficient of $x^2$, $2$ in this case, and the constant term of $-33$, which gives a value of $-66$.
• Use trial and error to find two numbers that multiply to give $-66$ but add up to the coefficient of $x$, which is $-5$.
• The two numbers are $-11$ and $6$.
• Rewrite the middle term as the sum of these two numbers $-5x = -11x+6x$.
  $$ 2x^2-5x-33=2x^2-11x + 6x-33=0 $$ Factor the largest common term from the first two terms, and also from the last two terms. $$ x(2x-11) + 3(2x-11)=0$$ $$(2x-11)(x + 3)=0 \quad \text{Factor $(2x-11)$} $$
• The roots are then obtained by solving $2x-11=0$ and $x+3=0$, which gives $\dfrac{11}{2}$ and $-3$ as the
  two roots of the quadratic equation $2x^2-5x-33=0$.

Completing a Square

Completing a square refers to the process of adding a constant term to a quadratic expression such that the result is a perfect square. Consider the expression $x^2+6x+4$, we can rewrite this as $$x^2+6x+9-9+4=x^2+6x+9-5$$ We further recognize that $x^2+6x+9=(x+3)^2$ is a perfect square, and therefore the original expression is equivalent to $(x+3)^2-5$.

In general if we are given a quadratic of the form $x^2+bx$, we add $\left(\dfrac{b}{2}\right)^2$ to the quadratic to obtain a perfect square. $$ x^2 + bx + \left(\dfrac{b}{2}\right)^2 = x^2+ bx + \dfrac{b^2}{4} = \left(x+ \dfrac{b}{2}\right)^2$$

Quadratic Expressions: Maximum and Minimum

Completing the square of a quadratic expression helps us to find the least and greatest value of such expressions. In general the quadratic expression $ax^2+bx+c$ attains its maxima ($a<0$) or minima ($a>0$) when $x = -\dfrac{b}{2a}$. The greatest and the least value of the quadratic expression can be found by substituting $x=-\dfrac{b}{2a}$ into the expression $ax^2+bx+c$.

Solved Example: $$ h(t) = -16t^2+24t+18$$
The equation above gives the height above ground $h$, in feet, of a ball $t$ seconds after it is launched vertically upward from a platform that is at a height of 18 feet. What is the greatest height the ball achieves?

1. $21$
2. $24$
3. $27$
4. $30$

Explanation:The easiest way to solve this problem is by recognizing that the maximum value of the expression occurs when $t = -\dfrac{b}{2a}= -\dfrac{24}{2(-16)} = \dfrac{3}{4}$, and we can find the maximum height by substituting $t=\dfrac{3}{4}$ in the expression $-16t^2+24t+18= -16\left(\dfrac{3}{4}\right)^2 + 24\left(\dfrac{3}{4}\right) + 18 = -9 + 18 + 18 = 27$.

We can also do this problem by completing the square, we first factor out the term $-16$ from the first two terms of the quadratic expression to obtain:
$$
\begin{align}
h(t) &= -16t^2+24t+18 && \text{Factor out $-16$}\\
&= -16\left[t^2 – \dfrac{3}{2}\right] + 18 && \text{Complete the square} \\
&=-16\left[t^2 – \dfrac{3}{2}+ \left(\dfrac{3}{4}\right)^2 – \left(\dfrac{3}{4}\right)^2\right] + 18 && \text{} \\
&= -16\left[t^2 – \dfrac{3}{2}+ \left(\dfrac{3}{4}\right)^2\right] +16\left(\dfrac{3}{4}\right)^2+ 18 && \text{}\\
& = -16\left(t-\dfrac{3}{4}\right)^2 + 27 && \\
\end{align}
$$
Note that the expression $\left(t-\dfrac{3}{4}\right)^2 \geq 0$ because it is a square of a number. This means that the term $-16\left(t-\dfrac{3}{4}\right)^2$ will always be less than or equal to zero. The maximum height will have a value of $27$ and this will happen when $t=\dfrac{3}{4}$.

Quadratic Formula

The quadratic formula can be used to obtain the solutions to the general quadratic equation $ax^2 + bx + c=0$:
$$ x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $$
Example: $2x^2-5x-33=0$, we have $a=2$, $b=-5$, and $c=-33$, replacing these values in the above quadratic formula yields $$ x = \dfrac{-(-5) \pm \sqrt{(-5)^2-4(2)(-33)}}{2(2)}=\dfrac{5 \pm \sqrt{289}}{4} = \dfrac{5 \pm 17}{4}$$ and the two solutions are $x=\dfrac{11}{2}$ and $x=-3$.

Solved Example: $$ \dfrac{x^2}{2} = mx + \dfrac{n}{2} $$
In the quadratic equation above, $m$ and $n$ are constants. What are the solutions for $x$ ?

1. $\quad m \pm \sqrt{m^2-n}$
2. $\quad m \pm \sqrt{m^2+n}$
3. $\quad -m \pm \sqrt{m^2-n}$
4. $\quad -m \pm \sqrt{m^2+n}$

Explanation: $$
\begin{align}
\dfrac{x^2}{2} &= mx + \dfrac{n}{2} && \text{Multiply both sides by 2}\\
x^2 &= 2mx +n && \text{Rewrite in standard form} \\
x^2 – 2mx – n &=0 && \text{Note $a=1$, $b=-2m$, $c=-n$} \\
x &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} && \text{Apply quadratic formula}\\
& = \dfrac{-(-2m) \pm \sqrt{(-2m)^2-4(1)(-n)}}{2(1)} && \\
& = \dfrac{2m \pm \sqrt{4m^2+4n}}{2} &&\\
& = \dfrac{2m \pm 2\sqrt{m^2+n}}{2} && \text{Divide throughout by 2}\\
& = m \pm \sqrt{m^2+n} &&\\
\end{align}
$$

Sum and Product of Roots

Let $r_1$ and $r_2$ be the roots of the quadratic equation $ax^2 + bx + c=0$, then the quadratic equation in the factored form can be written as $a(x-r_1)(x-r_2)=0$. If we multiply this expression and rearrange it, we obtain:
$$
\begin{align}
a(x-r_1)(x-r_2)&=0 &&\\
a(x^2 – xr_2 – xr_1 + r_1 r_2) &=0 &&\\
a[x^2 -x(r_1+r_2)+r_1r_2] &=0 &&\\
ax^2-xa(r_1+r_2)+ar_1r_2 &=0 &&\\
\end{align}
$$
This resulting expanded form is equivalent to the quadratic equation $ax^2 + bx + c=0$, and the coefficient of all the terms must be identical. If we equate the coefficient of $x$ term we obtain $-a(r_1+r_2) = b$, or in other words:
$$\textrm{Sum of Roots} = r_1 + r_2 = -\dfrac{b}{a} $$ Further if we equate the constant part of the two equations, we obtain $ar_1r_2=c$, or in other words: $$ \textrm{Product of Roots} = r_1 r_2 = \frac{c}{a} $$ The above two formulas are helpful in finding the sum and product of the roots without having to find the roots themselves.

Solved Example: $$ 2x^2=3x+2 $$ What is the sum of all the solutions to the quadratic equation above?

1. $\quad -\dfrac{5}{2}$
2. $\quad -\dfrac{3}{2}$
3. $\quad \quad \dfrac{3}{2}$
4. $\quad \quad \dfrac{5}{2}$

Explanation: Rewrite the equation in the standard quadratic form $ax^2+bx+c=0$ as $2x^2-3x-2=0$. We have $a=2$, $b=-3$, and $c=-2$. The sum of the roots is equal to $-\dfrac{b}{a}=-\dfrac{(-3)}{2} = \dfrac{3}{2}$. There is no need to waste time in first finding the two roots, which are $-\dfrac{1}{2}$ and $2$, and adding them to find their sum of $\dfrac{3}{2}$.

Nature of Roots

The table below summarizes how the discriminant $b^2-4ac$ determines the nature of the roots of a general quadratic equation $ax^2+bx+c=0$, and where the corresponding graph of the parabola $y=ax^2+bx+c=0$ falls with respect to the $x-$axis. I will review parabolas in a different article.
$$
\begin{array}{|c|c|c|}
\hline
\textbf{Condition} & \textbf{Nature of Roots} & \textbf{Graph of the}\\
& & \textbf{corresponding parabola} \\ \hline
b^2 – 4ac > 0 & \text{Roots are distinct and real} & \text{Parabola intersects the $x-$axis}\\
& & \text{at two distinct points}\\ \hline
b^2 – 4ac < 0 & \text{No real roots} & \text{Parabola does not }\\ & & \text{intersect the $x-$axis}\\ \hline b^2 - 4ac = 0 & \text{Roots are identical} & \text{Parabola is tangent} \\ & \text{also called a double root} & \text{to the $x-$axis}\\ \hline \end{array} $$ Example: If $b$ is a constant such that the equation $2x^2=bx-3$ has two distinct real solutions, which of the following could be the value of $b$ ?

1. $\quad -5$
2. $\quad -4$
3. $\quad -3$
4. $\quad 4$

Explanation: First write the quadratic equation in the standard form $2x^2-bx+3=0$. For the roots to be real and distinct $b^2-4ac>0$, which is equivalent to $(-b)^2 – 4(2)(3)>0$ or $b^2>24$. The only value in the answer choices that satisfies this condition is $-5$.

$$
\begin{array}{|c|c|c|}
\hline
\textbf{Rule} & \textbf{Arithmetic example} & \textbf{Algebraic example} \\ \hline
(a^m) (a^n) = a^{m+n} & (5^{3})(5^5) = 5^{3+5}=5^8 & (x^6)(x^{-4})=x^{6+(-4)} = x^2 \\ \hline
(a^m)^n = a^{mn} = (a^n)^m &(2^2)^3=2^{6}=64 &(3z^2)^3=(3^3)(z^2)^3=27z^{6} \\ \hline
a^{-n} = \dfrac{1}{a^n} \quad (a \neq 0)& 2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8} & x^{-3} = \dfrac{1}{x^3} \\ \hline
\dfrac{a^m}{a^n} = a^{m-n} & \dfrac{7^{8}}{7^5} = 7^{8-5}=7^3 & \dfrac{x^5}{x^{-4}}=x^{5-(-4)} = x^9\\ \hline
a^0=1 & (-5)^0=1 & x^0=1 \quad (x \neq 0) \\ \hline
(a \times b)^n = (a^n)(b^n) & (2\times 5)^6=(2^6)(5^6) =10^6 & (2x)^3 = (2)^3(x)^3= 8x^3\\ \hline
\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} & \left(\dfrac{3}{2}\right)^3 = \dfrac{3^3}{2^3}=\dfrac{27}{8} & \left(\dfrac{2x}{3y^{2}}\right)^3 = \dfrac{2^3 x^3}{3^{3} (y^{2})^3}=
\dfrac{8x^3}{27 y^{6}}\\ \hline
\end{array}
$$

Exponents: Algebraic Manipulation

You will be expected to rewrite and manipulate algebraic expressions containing exponent terms. Here I list some examples of common manipulations:
$$8^x = (2^3)^x = 2^{3x}$$ $$27^x = (3^3)^x = 3^{3x}$$ $$3^{6x} = (3^2)^{3x} = 9^{3x}$$ $$3^{6x} = (3^3)^{2x} = 27^{2x}$$ $$9(3^x) = (3^2)(3^x) = 3^{x+2} \quad \text{Note:} \quad 9(3^x) \neq 27^{x}$$ $$8(2^x) = (2^3)(2^x) = 2^{x+3} \quad \text{Note:} \quad 8(2^x) \neq 16^{x}$$ $$\dfrac{3^{x+1}}{3} =\dfrac{3^{x+1}}{3^1}= 3^{(x+1)-1} = 3^x \quad \text{Note:} \quad \dfrac{3^{x+1}}{3} \neq x+1$$ $$ \dfrac{9^x}{27^y} = \dfrac{(3^2)^x}{(3^3)^y} = \dfrac{3^{2x}}{3^{3y}} = 3^{2x-3y} $$

Exponents: Adding and Subtracting Terms

You will be asked to simplify exponent expressions such as $2^{8}+2^{8}$. The most common mistake students make is to add the exponents, $2^8 + 2^8 \neq 2^{16}$, instead $2^8 + 2^8 = 2^{9}$. There is no general exponent rule when adding powers of numbers that have the same base, however, there are cases where simplification is possible using other rules of arithmetic. If you see a question on the SAT that asks you to add exponent terms with the same bases, the best approach is to factor the largest common term which will almost always lead to simplification. For example:
$$2^8 + 2^8 = 2^8(1 + 1) = 2^8(2) = 2^8(2^1) = 2^{8+1} = 2^9$$

Additional Examples

$$ 2^4 + 2^4 + 2^4 + 2^4 = 2^4(1 + 1 + 1 + 1) = 2^4(4) = 2^4(2^2) = 2^{4+2} = 2^6 $$ $$ 2^{20} – 3(2^{18}) = 2^{18}(2^2 – 3) = 2^{18}(4-3) =2^{18} $$ $$ 27^{\frac{5}{3}} – 8(27) = 27(27^{\frac{5}{3} – 1} – 8) = 27(27^{\frac{2}{3}} – 8)
= 27[ (3^3)^{\frac{2}{3}} – 8] = 27( 3^2 – 8) = 27 $$ $$ 2^{x} + 2^{x+1} = 2^x + (2^{x})(2^1) = 2^{x}(1+2) =3(2^x)$$

Radicals and Fractional Exponents

• A square root of a number $n$ is a number that, when squared, is equal to $n$ or $(\sqrt{n})^2 = n$.
• An exponent of $\dfrac{1}{2}$ is the same as taking the square root of a number, $\sqrt{n} = n^{\frac{1}{2}}.$
• A cube root of a number $n$ is a number that, when cubed, is equal to $n$ or $(\sqrt[3]{n})^3 = n$.
• An exponent of $\dfrac{1}{3}$ is the same as taking the cube root of a number, $\sqrt[3]{n} = n^{\frac{1}{3}}$.

$$
\begin{array}{|c|c|}
\hline
\textbf{Rule} & \textbf{Example} \\ \hline
(\sqrt{a})^2 = a \quad (a>0) & (\sqrt{7})^2 = 7 \\ \hline
\sqrt{a^2} = a \quad (a>0) & \sqrt{7^2} = 7 \\ \hline
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} & \sqrt{6}\sqrt{8}=\sqrt{48} = \sqrt{(16)(3)} = \sqrt{16}\sqrt{3} = 4\sqrt{3} \\ \hline
\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} & \dfrac{\sqrt{6}}{\sqrt{24}} = \sqrt{\dfrac{6}{24}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2} \\ \hline
a^{\frac{m}{n}}= (a^{\frac{1}{n}})^m = (a^m)^{\frac{1}{n}}= (\sqrt[n]{a})^m=\sqrt[n]{a^m} & 27^{\frac{2}{3}} =
(27^2)^{\frac{1}{3}}=(27^{\frac{1}{3}})^2 = (\sqrt[3]{27})^2= (3)^2=9\\ \hline
\end{array}
$$

Here is a list of common exponent manipulations that you can expect on the SAT math section:
$$ \sqrt[6]{27} = 27^{\frac{1}{6}} = (3^3)^{\frac{1}{6}} = 3^{\frac{3}{6}} = 3^{\frac{1}{2}} = \sqrt{3}$$ $$ 64^{-\frac{2}{3}} = (2^6)^{-\frac{2}{3}} = 2^{6 \times -\frac{2}{3}} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16} $$ $$a^{\frac{3}{4}} = (a^{\frac{1}{4}})^3 = (\sqrt[4]{a})^3 = (a^3)^{\frac{1}{4}} = \sqrt[4]{a^3} $$ $$(\sqrt{x^{3}})^4 = [(x^{3})^{\frac{1}{2}}]^4 = x^{(3)(\frac{1}{2})(4)} = x^6$$ $$x \sqrt{x}=(x^1)(x^{\frac{1}{2}}) = x^{1+\frac{1}{2}} = x^{\frac{3}{2}}$$ $$ \sqrt[3]{8x^6} = (8x^6)^{\frac{1}{3}} = (8)^{\frac{1}{3}} (x^6)^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} (x)^{\frac{6}{3}} = 2x^2 $$ $$ \sqrt[6]{x^4 y^3 z^2} =(x^4 y^3 z^2)^{\frac{1}{6}} = x^{\frac{4}{6}} y^{\frac{3}{6}} z^{\frac{2}{6}} = x^{\frac{2}{3}} y^{\frac{1}{2}} z^{\frac{1}{3}} $$

Common Mistakes

Finally, I list the most common mistakes that students make when they have to simplify algebraic expressions with exponents.
$$
\begin{array}{|c|c|}
\hline
\textbf{Mistake} & \textbf{Example} \\ \hline
(3a)^3 \neq 3a^3 & \text{Instead} \quad (3a)^3 = (3^3)(a^3) = 27a^3 \\ \hline
\sqrt{a^2 + b^2} \neq a + b & \sqrt{3^2 + 4^2} \neq 3 + 4 \quad \text{instead} \quad \sqrt{3^2+4^2}=\sqrt{25}=5 \\ \hline
(a + b)^n \neq a^n + b^n & (2 + 3)^2 = 5^2 = 25 \neq 2^2 + 3^2 = 13 \\ \hline
(-a)^2 \neq -(a^2) & (-3)^2 = 9 \neq -(3^2) = -9 \quad \text{instead} \quad (-a)^2 = a^2 \\ \hline
\end{array}
$$