Solve 4x² + 7x - 2 = 0
The solutions of 4x² + 7x - 2 = 0 are x = 1/4 or x = -2.
Change a, b or c (whole numbers, a not 0) and the exact solutions follow.
How to work it out
- 1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = 7, c = -2.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is positive, so there are two different real solutions.
- 4
Method 1, factoring. The roots are rational, so the left side factors:
- 5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
- 6
Method 2, The quadratic formula works for every quadratic:
- 7
Simplify to get the solutions:
- 8
Check by putting x = 1/4 back in: a·x² + b·x + c = 4·(1/16) + 7·(1/4) + (-2) = 0 ✓
LaTeX source of the formulas on this page
\[ 4x^2 + 7x - 2 = 0\ \Rightarrow\ x = \frac{1}{4},\ -2 \]
\[ 4x^2 + 7x - 2 = 0 \]
\[ D = 7^2 - 4\cdot4\cdot(-2) = 49 + 32 = 81 \]
\[ \left(4x - 1\right)\left(x + 2\right) = 0 \]
\[ 4x - 1 = 0 \Rightarrow x = \frac{1}{4}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{81}}{8} \]
\[ x = \frac{1}{4}\quad\text{or}\quad x = -2 \]The parabola y = 4x² + 7x - 2
| x | y |
|---|---|
| -3.875 | 30.9375 |
| -2.875 | 10.9375 |
| -1.875 | -1.0625 |
| -0.875 | -5.0625 |
| 0.125 | -1.0625 |
| 1.125 | 10.9375 |
| 2.125 | 30.9375 |
Questions
What are the solutions of 4x² + 7x - 2 = 0?
The solutions of 4x² + 7x - 2 = 0 are x = 1/4 or x = -2.
Does 4x² + 7x - 2 = 0 factor?
Yes: (4x-1)(x+2) = 0.
What is the vertex of y = 4x² + 7x - 2?
(-7/8, -81/16). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -7/4, product = c/a = -1/2.