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Solve 4x² + 7x = 0

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

x² +x += 0
4x2+7x=0⇒x=0, −744x^2 + 7x = 0 \Rightarrow x = 0,\ -\frac{7}{4}
Discriminant D = 49 · x ≈ 0, -1.75

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

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Graph of y = 4x² + 7x-4-3-2-11210203040x = 0x = -1.75vertexy = 4x² + 7x
Graph of y = 4x² + 7x. The curve crosses the x-axis at x = 0 and x = -1.75; the vertex is at (-0.875, -3.0625).

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 = 7, c = 0.

    4x2+7x=04x^2 + 7x = 0
  2. 2

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

    D=72−4⋅4⋅0=49+0=49D = 7^2 - 4\cdot4\cdot0 = 49 + 0 = 49
  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)(4x+7)=0\left(x\right)\left(4x + 7\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=0⇒x=04x+7=0⇒x=−74x = 0 \Rightarrow x = 0\qquad 4x + 7 = 0 \Rightarrow x = -\frac{7}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±498x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{49}}{8}
  7. 7

    Simplify to get the solutions:

    x=0orx=−74x = 0\quad\text{or}\quad x = -\frac{7}{4}
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 + 7x = 0\ \Rightarrow\ x = 0,\ -\frac{7}{4} \]
\[ 4x^2 + 7x = 0 \]
\[ D = 7^2 - 4\cdot4\cdot0 = 49 + 0 = 49 \]
\[ \left(x\right)\left(4x + 7\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 4x + 7 = 0 \Rightarrow x = -\frac{7}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{49}}{8} \]
\[ x = 0\quad\text{or}\quad x = -\frac{7}{4} \]

The parabola y = 4x² + 7x

xy
-3.87532.9375
-2.87512.9375
-1.8750.9375
-0.875-3.0625
0.1250.9375
1.12512.9375
2.12532.9375

Questions

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

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

Does 4x² + 7x = 0 factor?

Yes: (x)(4x+7) = 0.

What is the vertex of y = 4x² + 7x?

(-7/8, -49/16). 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 = 0.

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