You can use the quadratic formula to find the solutions of any quadratic equation, once the equation is written in standard form.
The Quadratic Formula
\( x = \frac{- b \pm \sqrt{b^{2} - 4ac}}{2a} \)
Where a, b, and c are the values from the given quadratic equation, and x is the variable.
The answers obtained from using the quadratic formula can be:
Integers such as \( x = 3 \)
Radicals like \( x = \frac{5 \pm \sqrt{2}}{3} \)
Imaginary numbers as in \( x = \pm 10i \)
When solving quadratic equations, the words "roots" and "solutions" are sometimes used interchangeably. There's a slight technical difference. The root is the value of \( x \) that you get when solve the equation. The solution is the ordered pair \( \left( x,0 \right). \)
Think you got it?
Use the activity below to practice solving quadratic equations using the quadratic formula. Remember to write the quadratic equation in standard form and to identify the values of a, b, and c before using the formula. Be sure to check your answers.
Use the quadratic formula to find the roots of \( x^{2} - 7x + 12 = 0 \).
The roots are \( x = 4 \) and \( x = 3 \).
If you need help arriving at this answer, click the solution button.
Check to make sure the quadratic equation is given in standard form. |
The equation \( x^{2} - 7x + 12 = 0 \) matches the structure \( ax^{2} + bx + c = 0 \). The equation is in standard form. |
Read the values of a, b, and c from the quadratic equation. |
\( a = 1 \) \( b = - 7 \) \( c = 12 \) |
Substitute the values into the quadratic formula. Pay close attention to negative signs. |
\( x = \frac{- \left( - 7 \right) \pm \sqrt{\left( - 7 \right)^{2} - 4\left( 1 \right)\left( 12 \right)}}{2\left( 1 \right)} \) |
Simplify. Use the order of operations. |
\( x = \frac{7 \pm \sqrt{49 - 48}}{2} \) \( x = \frac{7 \pm \sqrt{1}}{2} \) \( x = \frac{7 \pm 1}{2} \) |
Solve. |
\( x = \frac{7 + 1}{2} = \frac{8}{2} = 4 \) and \( x = \frac{7 - 1}{2} = \frac{6}{2} = 3 \) |
Use the quadratic formula to find the roots of \( x^{2} + 4x + 4 = 0 \).
The root is \( x = - 2 \).
If you need help arriving at this answer, click the solution button.
Check to make sure the quadratic equation is given in standard form. |
The equation \( x^{2} + 4x + 4 = 0 \) is in standard form. |
Read the values of a, b, and c from the quadratic equation. |
\( a = 1 \) \( b = 4 \) \( c = 4 \) |
Substitute the values into the quadratic formula. Pay close attention to negative signs. |
\( x = \frac{- \left( 4 \right) \pm \sqrt{\left( 4 \right)^{2} - 4\left( 1 \right)\left( 4 \right)}}{2\left( 1 \right)} \) |
Simplify. Use the order of operations. |
\( x = \frac{- 4 \pm \sqrt{16 - 16}}{2} \) \( x = \frac{- 4 \pm \sqrt{0}}{2} \) \( x = \frac{- 4 \pm 0}{2} \) |
Solve. |
\( x = \frac{- 4 \pm 0}{2} = \frac{- 4}{2} = - 2 \) |
Use the quadratic formula to find the roots of \( 8x^{2} - 5x = - 3 \).
The roots are \( x = \frac{5 + i\sqrt{71}}{16} \) and \( x = \frac{5 - i\sqrt{71}}{16} \).
If you need help arriving at this answer, click the solution button.
Check to make sure the quadratic equation is given in standard form. |
The equation \( 8x^{2} - 5x = - 3 \) is not in standard form. Add 3 to both sides of the equation. The equation in standard form is \( 8x^{2} - 5x + 3 = 0 \). |
Read the values of a, b, and c from the quadratic equation. |
\( a = 8 \) \( b = - 5 \) \( c = 3 \) |
Substitute the values into the quadratic formula. Pay close attention to negative signs. |
\( x = \frac{- \left( - 5 \right) \pm \sqrt{\left( - 5 \right)^{2} - 4\left( 8 \right)\left( 3 \right)}}{2\left( 8 \right)} \) |
Simplify. Use the order of operations. |
\( x = \frac{5 \pm \sqrt{25 - 96}}{16} \) \( x = \frac{5 \pm \sqrt{- 71}}{16} \) Write the square root of the negative radicand as the product involving \( - 1 \). \( x = \frac{5 \pm \sqrt{\left( - 1 \right)(71)}}{16} \) The square root of \( - 1 = i \). \( x = \frac{5 \pm i\sqrt{\left( 71 \right)}}{16} \) |
Solve. |
\( x = \frac{5 + i\sqrt{71}}{16} \) and \( x = \frac{5 - i\sqrt{71}}{16} \) |
Use the quadratic formula to find the roots of \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \). Express your answers as decimals rounded to the nearest hundredth.
The roots are \( x = 10 - 2 \sqrt{15} \) and \( x = 10 + 2 \sqrt{15} \).
If you need help arriving at this answer, click the solution button.
Check to make sure the quadratic equation is given in standard form. |
The equation \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \) is in standard form. |
Read the values of a, b, and c from the quadratic equation. |
\( a = - \frac{1}{2} \) \( b = 10 \) \( c = - 20 \) |
Substitute the values into the quadratic formula. Pay close attention to negative signs. |
\( x = \frac{- (10) \pm \sqrt{10^{2} - 4\left( - \frac{1}{2} \right)( - 20)}}{2\left( - \frac{1}{2} \right)} \) |
Simplify. Use the order of operations. |
\( x = \frac{- 10 \pm \sqrt{100 - 40}}{2\left( - \frac{1}{2} \right)} \) \( x = \frac{- 10 \pm \sqrt{60}}{- 1} \) \( x = \frac{- 10 \pm 2 \sqrt{15}}{- 1} \) |
Solve. |
\( x = \frac{- 10 + 2 \sqrt{15}}{- 1} = 10 - 2 \sqrt{15} \) and \( x = \frac{- 10 - 2 \sqrt{15}}{- 1} = 10 + 2 \sqrt{15} \) |
Question
You discovered that the roots of the quadratic equation \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \) are \( x = 2.25 \) and \( x = 17.75 \). This equation represents the height of the roller coaster's first hill over time.
Use these solutions to determine how long it will take for the roller coaster's train cars to ascend and descend the first hill. The time unit is in seconds.
Find the difference between the two values. It is \( 17.75 - 2.25 = 15.5 \). It will take 15.5 seconds for the train cars to ascend and then descend the hill.
