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Solve 4x² + 12x = 0

The solutions of 4x² + 12x = 0 are x = 0 or x = -3.

x² +x += 0
4x2+12x=0⇒x=0, −34x^2 + 12x = 0 \Rightarrow x = 0,\ -3
Discriminant D = 144 · x ≈ 0, -3

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

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Graph of y = 4x² + 12x-5-4-3-2-112-10102030405060x = 0x = -3vertexy = 4x² + 12x
Graph of y = 4x² + 12x. The curve crosses the x-axis at x = 0 and x = -3; the vertex is at (-1.5, -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 = 12, c = 0.

    4x2+12x=04x^2 + 12x = 0
  2. 2

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

    D=122−4⋅4⋅0=144+0=144D = 12^2 - 4\cdot4\cdot0 = 144 + 0 = 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:

    4(x)(x+3)=04\left(x\right)\left(x + 3\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=0x+3=0⇒x=−3x = 0 \Rightarrow x = 0\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−12±1448x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{144}}{8}
  7. 7

    Simplify to get the solutions:

    x=0orx=−3x = 0\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 4·(0) + 12·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 12x = 0\ \Rightarrow\ x = 0,\ -3 \]
\[ 4x^2 + 12x = 0 \]
\[ D = 12^2 - 4\cdot4\cdot0 = 144 + 0 = 144 \]
\[ 4\left(x\right)\left(x + 3\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{144}}{8} \]
\[ x = 0\quad\text{or}\quad x = -3 \]

The parabola y = 4x² + 12x

xy
-4.527
-3.57
-2.5-5
-1.5-9
-0.5-5
0.57
1.527

Questions

What are the solutions of 4x² + 12x = 0?

The solutions of 4x² + 12x = 0 are x = 0 or x = -3.

Does 4x² + 12x = 0 factor?

Yes: 4(x)(x+3) = 0.

What is the vertex of y = 4x² + 12x?

(-3/2, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 0.

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