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Solve 5x² + 12x - 9 = 0

The solutions of 5x² + 12x - 9 = 0 are x = 3/5 or x = -3.

x² +x += 0
5x2+12x−9=0⇒x=35, −35x^2 + 12x - 9 = 0 \Rightarrow x = \frac{3}{5},\ -3
Discriminant D = 324 · x ≈ 0.6, -3

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

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Graph of y = 5x² + 12x - 9-5-4-3-2-1123-2020406080x = 0.6x = -3vertexy = 5x² + 12x - 9
Graph of y = 5x² + 12x - 9. The curve crosses the x-axis at x = 0.6 and x = -3; the vertex is at (-1.2, -16.2).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 12, c = -9.

    5x2+12x−9=05x^2 + 12x - 9 = 0
  2. 2

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

    D=122−4⋅5⋅(−9)=144+180=324D = 12^2 - 4\cdot5\cdot(-9) = 144 + 180 = 324
  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:

    (5x−3)(x+3)=0\left(5x - 3\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:

    5x−3=0⇒x=35x+3=0⇒x=−35x - 3 = 0 \Rightarrow x = \frac{3}{5}\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−12±32410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{324}}{10}
  7. 7

    Simplify to get the solutions:

    x=35orx=−3x = \frac{3}{5}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 3/5 back in: a·x² + b·x + c = 5·(9/25) + 12·(3/5) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 12x - 9 = 0\ \Rightarrow\ x = \frac{3}{5},\ -3 \]
\[ 5x^2 + 12x - 9 = 0 \]
\[ D = 12^2 - 4\cdot5\cdot(-9) = 144 + 180 = 324 \]
\[ \left(5x - 3\right)\left(x + 3\right) = 0 \]
\[ 5x - 3 = 0 \Rightarrow x = \frac{3}{5}\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{324}}{10} \]
\[ x = \frac{3}{5}\quad\text{or}\quad x = -3 \]

The parabola y = 5x² + 12x - 9

xy
-4.228.8
-3.23.8
-2.2-11.2
-1.2-16.2
-0.2-11.2
0.83.8
1.828.8

Questions

What are the solutions of 5x² + 12x - 9 = 0?

The solutions of 5x² + 12x - 9 = 0 are x = 3/5 or x = -3.

Does 5x² + 12x - 9 = 0 factor?

Yes: (5x-3)(x+3) = 0.

What is the vertex of y = 5x² + 12x - 9?

(-6/5, -81/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -12/5, product = c/a = -9/5.

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