THE QUADRATIC FORMULA AND DISCRIMINANT
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Quadratic Equation
Any equation that can be written in the form ax² + bx +c =0, a ≠ 0 is considered a quadratic equation. There are four methods for solving a quadratic equation. Solving Quadratic Equations
We will be focusing on method 4 on this page which is the quadratic formula. The neat thing about the quadratic formula is that it can be used to solve all types of quadratic equations regardless of their factorability or not. The quadratic formula can even be used to find imaginary and irrational roots of quadratic functions. The quadratic function is given by \[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] |
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The radicand portion of the quadratic formula \[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] (colored in red) is the discriminant and it can be used to describe the nature of the roots before you even find it. The roots can be one of the following cases.
1. Discriminant negative: Two imaginary distinct roots
2: Discriminant is zero: Double root
3. The discriminant is positive: Two distinct real roots, both rational or both irrational depending on if the discriminant is a perfect square (rational roots) or not.
1. Discriminant negative: Two imaginary distinct roots
2: Discriminant is zero: Double root
3. The discriminant is positive: Two distinct real roots, both rational or both irrational depending on if the discriminant is a perfect square (rational roots) or not.
The role of the discriminant is highlighted on the graphic to the left showing the three possible solutions you can have when solving quadratic equations.
STEPS FOR USING THE QUADRATIC FORMULA (HOW TO USE) STRATEGY GUIDE
- Write the equation in standard form. \[a{x^2} + bx + c = 0\]. If the equation is not in this form, use the properties of equality to place it in this form.
- Write down the quadratic formula \[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
- Find a, b, and c.
- Substitute the values of a, b, and c into the quadratic formula.
- Simplify using the order of operations.
Example of Solving using the Quadratic Formula
Solve \({x^2} - x - 12 = 0\) by using the quadratic formula
Set a=1, b=-1 and c=-12 since the equation is already in standard form. Substitute into the quadratic formula \[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] and that gives us \(\begin{array}{c}x = \frac{{ - ( - 1) \pm \sqrt {{{( - 1)}^2} - 4(1)( - 12)} }}{{2(1)}}\\ = \frac{{1 \pm \sqrt {49} }}{2}\\ = 4{\rm{ or }} - 3.\end{array}\)
Set a=1, b=-1 and c=-12 since the equation is already in standard form. Substitute into the quadratic formula \[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] and that gives us \(\begin{array}{c}x = \frac{{ - ( - 1) \pm \sqrt {{{( - 1)}^2} - 4(1)( - 12)} }}{{2(1)}}\\ = \frac{{1 \pm \sqrt {49} }}{2}\\ = 4{\rm{ or }} - 3.\end{array}\)