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Solve 4x² + 8x - 5 = 0

The solutions of 4x² + 8x - 5 = 0 are x = 1/2 or x = -5/2.

x² +x += 0
4x2+8x−5=0⇒x=12, −524x^2 + 8x - 5 = 0 \Rightarrow x = \frac{1}{2},\ -\frac{5}{2}
Discriminant D = 144 · x ≈ 0.5, -2.5

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

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Graph of y = 4x² + 8x - 5-5-4-3-2-1123-10102030405060x = 0.5x = -2.5vertexy = 4x² + 8x - 5
Graph of y = 4x² + 8x - 5. The curve crosses the x-axis at x = 0.5 and x = -2.5; the vertex is at (-1, -9).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = 8, c = -5.

    4x2+8x−5=04x^2 + 8x - 5 = 0
  2. 2

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

    D=82−4⋅4⋅(−5)=64+80=144D = 8^2 - 4\cdot4\cdot(-5) = 64 + 80 = 144
  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)(2x+5)=0\left(2x - 1\right)\left(2x + 5\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=122x+5=0⇒x=−522x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±1448x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{144}}{8}
  7. 7

    Simplify to get the solutions:

    x=12orx=−52x = \frac{1}{2}\quad\text{or}\quad x = -\frac{5}{2}
  8. 8

    Check by putting x = 1/2 back in: a·x² + b·x + c = 4·(1/4) + 8·(1/2) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 8x - 5 = 0\ \Rightarrow\ x = \frac{1}{2},\ -\frac{5}{2} \]
\[ 4x^2 + 8x - 5 = 0 \]
\[ D = 8^2 - 4\cdot4\cdot(-5) = 64 + 80 = 144 \]
\[ \left(2x - 1\right)\left(2x + 5\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{144}}{8} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = -\frac{5}{2} \]

The parabola y = 4x² + 8x - 5

xy
-427
-37
-2-5
-1-9
0-5
17
227

Questions

What are the solutions of 4x² + 8x - 5 = 0?

The solutions of 4x² + 8x - 5 = 0 are x = 1/2 or x = -5/2.

Does 4x² + 8x - 5 = 0 factor?

Yes: (2x-1)(2x+5) = 0.

What is the vertex of y = 4x² + 8x - 5?

(-1, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = -5/4.

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