10.2: Solving Quadratic Equations

Standard Form of A Quadratic Equation

In Chapter 5 we studied linear equations in one and two variables and methods for solving them. We observed that a linear equation in one variable was any equation that could be written in the form $ax + b = 0, a\not = 0$, and a linear equation in two variables was any equation that could be written in the form $ax + by = c$, where $a$ and $b$ are not both $0$. We now wish to study quadratic equations in one variable.

Quadratic Equation

A quadratic equation is an equation of the form $ax^2 + bx + c = 0, a \not = 0$.

The standard form of the quadratic equation is $ax^2 + bx + c = 0, a \not = 0$.

For a quadratic equation in standard form $ax^2 + bx + c = 0$,

$a$ is the coefficient of $x^2$.

$b$ is the coefficient of $x$.

$c$ is the constant term.

Sample Set A

The following are quadratic equations.

Example $\PageIndex{1}$

$3x^2 + 2x - 1 = 0$. $a = 3, b = 2, c = -1$

Example $\PageIndex{2}$

$5x^2 + 8x = 0$. $a = 5, b = 8, c = 0$

Notice that this equation could be written $5x^2 + 8x + 0 = 0$. Now it is clear that $c = 0$.

Example $\PageIndex{3}$

$x^2 + 7 = 0$. $a = 1, b = 0, c = 7$.

Notice that this equation could be written $x^2 + 0x + 7 = 0$. Now it is clear that $b = 0$

The following are not quadratic equations.

Example $\PageIndex{4}$

$3x + 2 = 0$. $a = 0$. This equation is linear.

Example $\PageIndex{5}$

$8x^2 + \dfrac{3}{x} - 5 = 0$

The expression on the left side of the equal sign has a variable in the denominator and, therefore is not a quadratic.

Practice Set A

Which of the following equations are quadratic equations? Answer “yes” or “no” to each equation.

Practice Problem $\PageIndex{1}$

$6x^2 - 4x + 9 = 0$

Answer

yes

Practice Problem $\PageIndex{2}$

$5x+8=0$

Answer

no

Practice Problem $\PageIndex{3}$

$4x^3 - 5x^2 + x + 6 = 8$

Answer

no

Practice Problem $\PageIndex{4}$

$4x^2 - 2x + 4 = 1$

Answer

yes

Practice Problem $\PageIndex{5}$

$\dfrac{2}{x} - 5x^2 = 6x + 4$

Answer

no

Practice Problem $\PageIndex{6}$

$9x^2 - 2x + 6 = 4x^2 + 8$

Answer

yes

Zero-Factor Property

Our goal is to solve quadratic equations. The method for solving quadratic equations is based on the zero-factor property of real numbers. We were introduced to the zero-factor property in Section 8.2. We state it again.

Zero-Factor Property

If two numbers $a$ and $b$ are multiplied together adn the resulting product is $0$, then at least one of the numbers must be $0$. Algebraically, if $a \cdot b = 0$, then $a = 0$ or both $a = 0$ and $b = 0$.

Sample Set B

Use the zero-factor property to solve each equation.

Example $\PageIndex{6}$

If $9x = 0$, then $x$ must be $0$.

Example $\PageIndex{7}$

If $-2x^2 = 0$, then $x^2 = 0, x = 0$

Example $\PageIndex{8}$

If $5$ then $x-1$ must be $0$, since $5$ is not zero.

$\begin{array}{flushleft} x - 1 &= 0\\ x &= 1 \end{array}$

Example $\PageIndex{9}$

If $x(x+6) = 0$, then

$\begin{array}{flushleft} x &= 0 & \text{ or } & x+6&=0\\ x&=0, -6 && x &= -6 \end{array}$

Example $\PageIndex{10}$

If $(x+2)(x+3) = 0$, then

$\begin{array}{flushleft} x + 2 &= 0 & \text{ or } & x + 3 &= 0\\ x &= -2 && x &= -3\\ x &= -2, -3 \end{array}$

Example $\PageIndex{11}$

If $(x+10)(4x - 5) = 0$, then

$\begin{array}{flushleft} x + 10 &= 0 & \text{ or } & 4x - 5 &= 0\\ x &= -10 && 4x &= 5\\ x &= -10, \dfrac{5}{4} && x &= \dfrac{5}{4} \end{array}$

Practice Set B

Use the zero-factor property to solve each equation.

Practice Problem $\PageIndex{7}$

$6(a−4)=0$

Answer

$a=4$

Practice Problem $\PageIndex{8}$

$(y+6)(y−7)=0$

Answer

$y=−6, 7$

Practice Problem $\PageIndex{9}$

$(x+5)(3x−4)=0$

Answer

$x = -5, \dfrac{4}{3}$

Exercises

For the following problems, write the values of $a$, $b$, and $c$ in quadratic equations.

Exercise $\PageIndex{1}$

$3x^2 + 4x - 7 = 0$

Answer

$3,4,−7$

Exercise $\PageIndex{2}$

$7x^2 + 2x + 8 = 0$

Exercise $\PageIndex{3}$

$2y^2 - 5y + 5 = 0$

Answer

$2,−5,5$

Exercise $\PageIndex{4}$

$7a^2 + a - 8 = 0$.

Exercise $\PageIndex{5}$

$-3a^2 + 4a - 1 = 0$

Answer

$−3,4,−1$

Exercise $\PageIndex{6}$

$7b^2 + 3b + 0$

Exercise $\PageIndex{7}$

$2x^2 + 5x + 0$

Answer

$2, 5, 0$

Exercise $\PageIndex{8}$

$4y^2 + 9 = 0$

Exercise $\PageIndex{9}$

$8a^2 - 2a = 0$

Answer

$8,−2,0$

Exercise $\PageIndex{10}$

$6x^2 = 0$

Exercise $\PageIndex{11}$

$4y^2 = 0$

Answer

$4, 0, 0$

Exercise $\PageIndex{12}$

$5x^2 - 3x + 9 = 4x^2$

Exercise $\PageIndex{13}$

$7x^2 + 2x + 1 = 6x^2 + x - 9$

Answer

$1, 1, 10$

Exercise $\PageIndex{14}$

$-3x^2 + 4x - 1 = -4x^2 - 4x + 12$

Exercise $\PageIndex{15}$

$5x - 7 = -3x^2$

Answer

$3,5,−7$

Exercise $\PageIndex{16}$

$3x - 7 = -2x^2 + 5x$

Exercise $\PageIndex{17}$

$0 = x^2 + 6x - 1$

Answer

$1,6,−1$

Exercise $\PageIndex{18}$

$9 = x^2$

Exercise $\PageIndex{19}$

$x^2 = 9$

Answer

$1,0,−9$

Exercise $\PageIndex{20}$

$0 = -x ^2$

For the following problems, use the zero-factor property to solve the equations.

Exercise $\PageIndex{21}$

$4x = 0$

Answer

$x=0$

Exercise $\PageIndex{22}$

$16y=0$

Exercise $\PageIndex{23}$

$9a=0$

Answer

$a=0$

Exercise $\PageIndex{24}$

$4m=0$

Exercise $\PageIndex{25}$

$3(k+7)=0$

Answer

$k=−7$

Exercise $\PageIndex{26}$

$8(y−6)=0$

Exercise $\PageIndex{27}$

$−5(x+4)=0$

Answer

$x=−4$

Exercise $\PageIndex{28}$

$−6(n+15)=0$

Exercise $\PageIndex{29}$

$y(y−1)=0$

Answer

$y=0,1$

Exercise $\PageIndex{30}$

$a(a−6)=0$

Exercise $\PageIndex{31}$

$n(n+4)=0$

Answer

$n=0,−4$

Exercise $\PageIndex{32}$

$x(x+8)=0$

Exercise $\PageIndex{33}$

$9(a−4)=0$

Answer

$a=4$

Exercise $\PageIndex{34}$

$−2(m+11)=0$

Exercise $\PageIndex{35}$

$x(x+7) = 0$

Answer

$x=−7 \text{ or } x=0$

Exercise $\PageIndex{36}$

$n(n−10)=0$

Exercise $\PageIndex{37}$

$(y−4)(y−8)=0$

Answer

$y=4 \text{ or } y=8$

Exercise $\PageIndex{38}$

$(k−1)(k−6)=0$

Exercise $\PageIndex{39}$

$(x+5)(x+4)=0$

Answer

$x=−4 \text{ or } x=−5$

Exercise $\PageIndex{40}$

$(y+6)(2y+1)=0$

Exercise $\PageIndex{41}$

$(x−3)(5x−6)=0$

Answer

$x = \dfrac{6}{5} \text{ or } x = 3$

Exercise $\PageIndex{42}$

$(5a+1)(2a−3)=0$

Exercise $\PageIndex{43}$

$(6m+5)(11m−6)=0$

Answer

$m = -\dfrac{5}{6} \text{ or } m = \dfrac{6}{11}$

Exercise $\PageIndex{44}$

$(2m−1)(3m+8)=0$

Exercise $\PageIndex{45}$

$(4x+5)(2x−7)=0$

Answer

$x = \dfrac{-5}{4}, \dfrac{7}{2}$

Exercise $\PageIndex{46}$

$(3y + 1)(2y + 1) = 0$

Exercise $\PageIndex{47}$

$(7a + 6)(7a - 6) = 0$

Answer

$a = \dfrac{-6}{7}, \dfrac{6}{7}$

Exercise $\PageIndex{48}$

$(8x+11)(2x−7)=0$

Exercise $\PageIndex{49}$

$(5x−14)(3x+10)=0$

Answer

$x = \dfrac{14}{5}, \dfrac{-10}{3}$

Exercise $\PageIndex{50}$

$(3x−1)(3x−1)=0$

Exercise $\PageIndex{51}$

$(2y+5)(2y+5)=0$

Answer

$y = \dfrac{-5}{2}$

Exercise $\PageIndex{52}$

$(7a - 2)^2 = 0$

Exercise $\PageIndex{53}$

$(5m - 6)^2 = 0$

Answer

$m = \dfrac{6}{5}$

Exercises For Review

Exercise $\PageIndex{54}$

Factor $12ax - 3x + 8a - 2$ by grouping.

Exercise $\PageIndex{55}$

Construct the graph of $6x + 10y - 60 = 0$

Answer

Exercise $\PageIndex{56}$

Find the difference: $\dfrac{1}{x^2 + 2x + 1} - \dfrac{1}{x^2 - 1}$.

Exercise $\PageIndex{57}$

Simplify $\sqrt{7}(\sqrt{2} + 2)$

Answer

$\sqrt{14} + 2\sqrt{7}$

Exercise $\PageIndex{58}$

Solve the radical equation $\sqrt{3x + 10} = x + 4$