WordGPT

Solve 2x² + 10x - 9 = 0

The solutions of 2x² + 10x - 9 = 0 are x = (-5 + √43)/2 or x = (-5 - √43)/2, that is x ≈ 0.77871926 and x ≈ -5.7787193.

x² +x += 0
2x2+10x−9=0⇒x=−5+432, −5−4322x^2 + 10x - 9 = 0 \Rightarrow x = \frac{-5 + \sqrt{43}}{2},\ \frac{-5 - \sqrt{43}}{2}
Discriminant D = 172 · x ≈ 0.778719, -5.77872

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

Explain it step by step in chatMake a short video of it
Graph of y = 2x² + 10x - 9-8-6-4-22-30-20-101020304050x = 0.77871926x = -5.7787193vertexy = 2x² + 10x - 9
Graph of y = 2x² + 10x - 9. The curve crosses the x-axis at x = 0.77871926 and x = -5.7787193; the vertex is at (-2.5, -21.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = 10, c = -9.

    2x2+10x−9=02x^2 + 10x - 9 = 0
  2. 2

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

    D=102−4⋅2⋅(−9)=100+72=172D = 10^2 - 4\cdot2\cdot(-9) = 100 + 72 = 172
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±1724x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{172}}{4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−5+432orx=−5−432x = \frac{-5 + \sqrt{43}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{43}}{2}
  6. 6

    As decimals: x ≈ 0.77871926 and x ≈ -5.7787193.

  7. 7

    Check numerically: for x ≈ 0.77871926 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 10x - 9 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{43}}{2},\ \frac{-5 - \sqrt{43}}{2} \]
\[ 2x^2 + 10x - 9 = 0 \]
\[ D = 10^2 - 4\cdot2\cdot(-9) = 100 + 72 = 172 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{172}}{4} \]
\[ x = \frac{-5 + \sqrt{43}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{43}}{2} \]

The parabola y = 2x² + 10x - 9

xy
-5.5-3.5
-4.5-13.5
-3.5-19.5
-2.5-21.5
-1.5-19.5
-0.5-13.5
0.5-3.5

Questions

What are the solutions of 2x² + 10x - 9 = 0?

The solutions of 2x² + 10x - 9 = 0 are x = (-5 + √43)/2 or x = (-5 - √43)/2, that is x ≈ 0.77871926 and x ≈ -5.7787193.

Does 2x² + 10x - 9 = 0 factor?

Not over the integers, because the discriminant 172 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 10x - 9?

(-5/2, -43/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = -9/2.

Similar equations

    Solve 2x² + 10x - 9 = 0: x = (-5 + √43)/2 or x = (-5 - √43)/2 (steps) | WordGPT