Quadratic Formula
The quadratic formula is used to find, roots, or zeroes, to quadratic functions when the equation isn't factorable and solving for x when y = 0 is too difficult. This formula also gives the x-value of the vertex and the discriminant provides the number of solutions.
For any quadratic equation of the form y = ax2 + bx + c, the quadratic formula below
will find the roots, or zeroes, of the equation. The roots of a quadratic function are the same as its zeroes. They are where the graph crosses the x-axis, or simply put, where y = 0. A quadratic function can have 0, 1, or 2 roots.
Example 1:
y = 4x2 - 6x + 7
This problem cannot be factored and there is no easy way to solve for x when y = 0. So we must use the quadratic formula.
Step 1: First we find a, b, and c.
y = 4x2 - 6x + 7a b c
This equation is already written in the form of y = ax2 + bx + c so we have a = 4, b = -6, and c = 7.
Step 2 : Now we substitute these values into the formula and use the order of operations to simplify
| Quadratic Formula | |
| Substitute 4,-6, and 7 for a, b, and c, respectively. | |
| -(-6) can be simplified to 6. | |
| (-6)2 = (-6)(-6) = 36. | |
| 4(4)(7) is equal to 112 and multiplying 2 and 4 equals 8. | |
| 36 – 112 = -76 which is negative. Square roots of negative numbers are not possible in the set of real numbers, so we have no solution. |
Answer : no roots
Example 2:
y = - 16x + x2 -7
Rewriting this so that it is in the form of y = ax2 + bx + c , we get
y = x2 – 16x - 7 We get a = 1, b = -16, c = -7.
a b c
| Substitute 1,-16, and -7 into the formula. | |
| -(-16) simplifies to 16. | |
| (-16)2 = (-16)(-16) = 256. | |
| Multiplying 4(1)(-7) equals 28 and 2 (1)= 2. | |
| 256+28 = 284 | |
| Simplify the radical as much as possible |
Answer:
| Divide numerator and denominator by 2 to simplify | |
| The two roots are and |
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Related Links: Quadratic Functions Solving Quadratic Equations Quiz Quadratic Function Standard Form Quadratic Function Vertex Form |