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Quadratic Formula

2.3 Quadratic formula (EMBFK)

It is not always possible to solve a quadratic equation by factorisation and it can take a long time to complete the square. The method of completing the square provides a way to derive a formula that can be used to solve any quadratic equation. The quadratic formula provides an easy and fast way to solve quadratic equations.

Consider the standard form of the quadratic equation \(ax^2 + bx + c = 0\). Divide both sides by \(a\) \((a \ne 0)\) to get

\[x^2 + \frac{bx}{a} + \frac{c}{a} = 0\]

Now using the method of completing the square, we must halve the coefficient of \(x\) and square it. We then add and subtract \(\left( \dfrac{b}{2a} \right)^2\) so that the equation remains true.

\begin{align*} x^2 + \frac{bx}{a} + \frac{b^2}{4a^2} -\frac{b^2}{4a^2} + \frac{c}{a} &= 0 \\ \left( x^2 + \frac{bx}{a} + \frac{b^2}{4a^2} \right) -\frac{b^2}{4a^2} + \frac{c}{a} &= 0 \\ \left( x + \frac{b}{2a} \right)^2 -\frac{b^2-4ac}{4a^2} &= 0 \end{align*}

We add the constant to both sides and take the square root of both sides of the equation, being careful to include a positive and negative answer.

\begin{align*} \left( x + \frac{b}{2a} \right)^2 &= \frac{b^2-4ac}{4a^2} \\ \sqrt{\left( x + \frac{b}{2a} \right)^2} &= \pm \sqrt{\frac{b^2-4ac}{4a^2}} \\ x + \frac{b}{2a} &= \pm \dfrac{\sqrt{b^2-4ac}}{2a} \\ x &= - \dfrac{b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a} \\ x &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \end{align*}

Therefore, for any quadratic equation \(ax^2+bx+c = 0\) we can determine two roots \[x = \dfrac{-b + \sqrt{b^2-4ac}}{2a} \text{ or } x = \dfrac{-b - \sqrt{b^2-4ac}}{2a}\]

It is important to notice that the expression \({b}^{2}-4ac\) must be greater than or equal to zero for the roots of the quadratic to be real. If the expression under the square root sign is less than zero, then the roots are non-real (imaginary).

Worked example 8: Using the quadratic formula

Solve for \(x\) and leave your answer in simplest surd form: \(2x^2+3x=7\)

Check whether the expression can be factorised

The expression cannot be factorised, so the general quadratic formula must be used.

Write the equation in the standard form \(a{x}^{2}+bx+c=0\)

\[2x^2+3x-7=0\]

Identify the coefficients to substitute into the formula

\[a = 2; \qquad b = 3; \qquad c = -7\]

Apply the quadratic formula

Always write down the formula first and then substitute the values of \(a\), \(b\) and \(c\).

\begin{align*} x &= \frac{-b±\sqrt{{b}^{2}-4ac}}{2a} \\ &= \frac{-\left(3\right)±\sqrt{{\left(3\right)}^{2}-4\left(2\right)\left(-7\right)}}{2\left(2\right)} \\ &= \frac{-3±\sqrt{65}}{4} \end{align*}

Write the final answer

The two roots are \(x=\dfrac{-3+\sqrt{65}}{4}\) or \(x=\dfrac{-3-\sqrt{65}}{4}\).

Worked example 9: Using the quadratic formula

Find the roots of the function \(f(x)= x^2-5x+8\).

Finding the roots

To determine the roots of \(f(x)\), we let \(x^2-5x+8 = 0\).

Check whether the expression can be factorised

The expression cannot be factorised, so the general quadratic formula must be used.

Identify the coefficients to substitute into the formula

\[a = 1; \qquad b = -5; \qquad c = 8\]

Apply the quadratic formula

\begin{align*} x &= \frac{-b±\sqrt{{b}^{2}-4ac}}{2a} \\ &= \frac{-\left(-5\right)±\sqrt{{\left(-5\right)}^{2}-4\left(1\right)\left(8\right)}}{2\left(1\right)} \\ &= \frac{5±\sqrt{-7}}{2} \end{align*}

Write the final answer

There are no real roots for \(f(x)= x^2-5x+8\) since the expression under the square root is negative (\(\sqrt{-7}\) is not a real number). This means that the graph of the quadratic function has no \(x\)-intercepts; the entire graph lies above the \(x\)-axis.

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Solution by the quadratic formula

Exercise 2.3

Solve the following using the quadratic formula.

\(3{t}^{2}+t-4=0\)

\begin{align*} 3t^2 + t - 4 &= 0\\ t &= \dfrac{-(1) \pm \sqrt{1^2 - 4(3)(-4)}}{2(3)}\\ & = \dfrac{-1 \pm \sqrt{1+48}}{6}\\ &=\dfrac{-1 \pm \sqrt{49}}{6}\\ &=\dfrac{-1 \pm 7}{6}\\ t = \dfrac{-1+7}{6}= \frac{6}{6}=1 &\text{ or } t = \dfrac{-1-7}{6}=\frac{-8}{6} = \frac{-4}{3} \end{align*}

\({x}^{2}-5x-3=0\)

\begin{align*} x^2 - 5x - 3 &= 0\\ x &= \dfrac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-3)}}{2(1)}\\ &= \dfrac{5 \pm \sqrt{25 + 12}}{2}\\ &=\dfrac{5 \pm \sqrt{37}}{2}\\ \text{therefore } x =\dfrac{5 + \sqrt{37}}{2} &\text{ or } x =\dfrac{5 - \sqrt{37}}{2} \end{align*}

\(2{t}^{2}+6t+5=0\)

\begin{align*} 2t^2 + 6t + 5 &= 0\\ t&=\dfrac{-6 \pm \sqrt{(6)^2 - 4(2)(5)}}{2(2)}\\ &= \dfrac{-6 \pm \sqrt{36 - 40}}{4}\\ &=\dfrac{-6 \pm \sqrt{-4}}{4}\\ \text{No real solution} \end{align*}

\(2p(2p+1)=2\)

\begin{align*} 4p^2 + 2p - 2 &= 0\\ 2p^2 + p - 1 &= 0\\ p &= \dfrac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)}\\ &= \dfrac{-1 \pm \sqrt{1+8}}{4}\\ &= \dfrac{-1 \pm \sqrt{9}}{4} \\ &= \dfrac{-1 \pm 3}{4} \\ \text{therefore } p =\dfrac{-1 + 3}{4} = \frac{1}{2} &\text{ or } p =\dfrac{-1 - 3}{4} = -1 \end{align*}

\(-3{t}^{2}+5t-8=0\)

\begin{align*} -3t^2 + 5t -8 &= 0\\ t&= \dfrac{-5 \pm \sqrt{5^2 -4(-3)(-8)}}{2(-3)}\\ &= \dfrac{-5 \pm \sqrt{25-96}}{-6}\\ &=\dfrac{-5 \pm \sqrt{-71}}{-6}\\ \text{No real solution} \end{align*}

\(5{t}^{2}+3t-3=0\)

\begin{align*} 5t^2 + 3t - 3 &= 0\\ t &= \dfrac{-3 \pm \sqrt{3^2 - 4(5)(-3)}}{2(5)}\\ &= \dfrac{-3 \pm \sqrt{9+60}}{10}\\ &=\dfrac{-3 \pm \sqrt{69}}{10}\\ \text{therefore } t =\dfrac{-3 + \sqrt{69}}{10} &\text{ or } t =\dfrac{-3 - \sqrt{69}}{10} \end{align*}

\({t}^{2}-4t+2=0\)

\begin{align*} t^2 - 4t + 2 &= 0\\ t &= \dfrac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(2)}}{2(1)}\\ &= \dfrac{4 \pm \sqrt{16-8}}{2}\\ &=\dfrac{4 \pm \sqrt{8}}{2}\\ &= 2 \pm \sqrt{2} \end{align*}

\(9(k^2 - 1) = 7k\)

\begin{align*} 9k^2 - 7k - 9 &= 0\\ k &= \dfrac{-(-7) \pm \sqrt{(-7)^2 - 4(9)(-9)}}{2(9)}\\ &= \dfrac{7 \pm \sqrt{49 + 324}}{18}\\ &=\dfrac{7 \pm \sqrt{373}}{18}\\ k = \dfrac{7 + \sqrt{373}}{18} &\text{ or } k =\dfrac{7 - \sqrt{373}}{18} \end{align*}

\(3f -2 = -2f^2\)

\begin{align*} 2f^2 + 3f - 2 &= 0\\ f &= \dfrac{-3 \pm \sqrt{3^2-4(2)(-2)}}{2(2)}\\ &= \dfrac{-3 \pm \sqrt{9+16}}{4}\\ &= \dfrac{-3 \pm \sqrt{25}}{4}\\ &= \dfrac{-3 \pm 5}{4}\\ \text{therefore } f =\dfrac{-3 + 5}{4} = \frac{1}{2}&\text{ or } f =\dfrac{-3 - 5}{4} = -2 \end{align*}

\({t}^{2}+t+1=0\)

\begin{align*} t^2 + t +1 &= 0\\ t&= \dfrac{-1 \pm \sqrt{1-4(1)(1)}}{2(1)}\\ &= \dfrac{-1 \pm \sqrt{-3}}{2}\\ \text{No real solution} \end{align*}