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Solve -3x² - 7x + 10 = 0

The solutions of -3x² - 7x + 10 = 0 are x = -10/3 or x = 1.

x² +x += 0
−3x2−7x+10=0⇒x=−103, 1-3x^2 - 7x + 10 = 0 \Rightarrow x = -\frac{10}{3},\ 1
Discriminant D = 169 · x ≈ -3.33333, 1

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

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Graph of y = -3x² - 7x + 10-5-4-3-2-1123-50-40-30-20-101020x = -3.3333333x = 1vertexy = -3x² - 7x + 10
Graph of y = -3x² - 7x + 10. The curve crosses the x-axis at x = -3.3333333 and x = 1; the vertex is at (-1.1666667, 14.083333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -7, c = 10.

    −3x2−7x+10=0-3x^2 - 7x + 10 = 0
  2. 2

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

    D=(−7)2−4⋅(−3)⋅10=49+120=169D = (-7)^2 - 4\cdot(-3)\cdot10 = 49 + 120 = 169
  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:

    −(3x+10)(x−1)=0-\left(3x + 10\right)\left(x - 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:

    3x+10=0⇒x=−103x−1=0⇒x=13x + 10 = 0 \Rightarrow x = -\frac{10}{3}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±169−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{169}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−103orx=1x = -\frac{10}{3}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -10/3 back in: a·x² + b·x + c = -3·(100/9) + (-7)·(-10/3) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 7x + 10 = 0\ \Rightarrow\ x = -\frac{10}{3},\ 1 \]
\[ -3x^2 - 7x + 10 = 0 \]
\[ D = (-7)^2 - 4\cdot(-3)\cdot10 = 49 + 120 = 169 \]
\[ -\left(3x + 10\right)\left(x - 1\right) = 0 \]
\[ 3x + 10 = 0 \Rightarrow x = -\frac{10}{3}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{169}}{-6} \]
\[ x = -\frac{10}{3}\quad\text{or}\quad x = 1 \]

The parabola y = -3x² - 7x + 10

xy
-4.1667-12.917267
-3.16672.0829333
-2.166711.083133
-1.166714.083333
-0.166711.083533
0.83332.0837333
1.8333-12.916067

Questions

What are the solutions of -3x² - 7x + 10 = 0?

The solutions of -3x² - 7x + 10 = 0 are x = -10/3 or x = 1.

Does -3x² - 7x + 10 = 0 factor?

Yes: -1(3x+10)(x-1) = 0.

What is the vertex of y = -3x² - 7x + 10?

(-7/6, 169/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -7/3, product = c/a = -10/3.

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