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10.2: Solving Quadratic Equations

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    49402
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    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 \(6x + 10y - 60 = 0\)

    An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

    Answer

    A graph of a line passing through two points coordinates zero, six and five, three.

    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 \(\sqrt{3x + 10} = x + 4\)


    This page titled 10.2: Solving Quadratic Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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