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Solve 2x² + 7x - 4 = 0

The solutions of 2x² + 7x - 4 = 0 are x = 1/2 or x = -4.

x² +x += 0
2x2+7x−4=0⇒x=12, −42x^2 + 7x - 4 = 0 \Rightarrow x = \frac{1}{2},\ -4
Discriminant D = 81 · x ≈ 0.5, -4

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = 2x² + 7x - 4-6-5-4-3-2-1123-10102030x = 0.5x = -4vertexy = 2x² + 7x - 4
Graph of y = 2x² + 7x - 4. The curve crosses the x-axis at x = 0.5 and x = -4; the vertex is at (-1.75, -10.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = 7, c = -4.

    2x2+7x−4=02x^2 + 7x - 4 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=72−4⋅2⋅(−4)=49+32=81D = 7^2 - 4\cdot2\cdot(-4) = 49 + 32 = 81
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (2x−1)(x+4)=0\left(2x - 1\right)\left(x + 4\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x−1=0⇒x=12x+4=0⇒x=−42x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 4 = 0 \Rightarrow x = -4
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±814x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{81}}{4}
  7. 7

    Simplify to get the solutions:

    x=12orx=−4x = \frac{1}{2}\quad\text{or}\quad x = -4
  8. 8

    Check by putting x = 1/2 back in: a·x² + b·x + c = 2·(1/4) + 7·(1/2) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 7x - 4 = 0\ \Rightarrow\ x = \frac{1}{2},\ -4 \]
\[ 2x^2 + 7x - 4 = 0 \]
\[ D = 7^2 - 4\cdot2\cdot(-4) = 49 + 32 = 81 \]
\[ \left(2x - 1\right)\left(x + 4\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 4 = 0 \Rightarrow x = -4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{81}}{4} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = -4 \]

The parabola y = 2x² + 7x - 4

xy
-4.757.875
-3.75-2.125
-2.75-8.125
-1.75-10.125
-0.75-8.125
0.25-2.125
1.257.875

Questions

What are the solutions of 2x² + 7x - 4 = 0?

The solutions of 2x² + 7x - 4 = 0 are x = 1/2 or x = -4.

Does 2x² + 7x - 4 = 0 factor?

Yes: (2x-1)(x+4) = 0.

What is the vertex of y = 2x² + 7x - 4?

(-7/4, -81/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/2, product = c/a = -2.

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