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Solve 2x² - 7x - 4 = 0

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

x² +x += 0
2x2−7x−4=0⇒x=4, −122x^2 - 7x - 4 = 0 \Rightarrow x = 4,\ -\frac{1}{2}
Discriminant D = 81 · x ≈ 4, -0.5

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

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Graph of y = 2x² - 7x - 4-3-2-1123456-10102030x = 4x = -0.5vertexy = 2x² - 7x - 4
Graph of y = 2x² - 7x - 4. The curve crosses the x-axis at x = 4 and x = -0.5; 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=(−7)2−4⋅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:

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

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

    x−4=0⇒x=42x+1=0⇒x=−12x - 4 = 0 \Rightarrow x = 4\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  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=4orx=−12x = 4\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 2·(16) + (-7)·(4) + (-4) = 0 ✓

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

The parabola y = 2x² - 7x - 4

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

Questions

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

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

Does 2x² - 7x - 4 = 0 factor?

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