WordGPT

Solve 4x² - 9x - 9 = 0

The solutions of 4x² - 9x - 9 = 0 are x = 3 or x = -3/4.

x² +x += 0
4x2−9x−9=0⇒x=3, −344x^2 - 9x - 9 = 0 \Rightarrow x = 3,\ -\frac{3}{4}
Discriminant D = 225 · x ≈ 3, -0.75

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 = 4x² - 9x - 9-3-2-112345-20204060x = 3x = -0.75vertexy = 4x² - 9x - 9
Graph of y = 4x² - 9x - 9. The curve crosses the x-axis at x = 3 and x = -0.75; the vertex is at (1.125, -14.0625).

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 = -9, c = -9.

    4x2−9x−9=04x^2 - 9x - 9 = 0
  2. 2

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

    D=(−9)2−4⋅4⋅(−9)=81+144=225D = (-9)^2 - 4\cdot4\cdot(-9) = 81 + 144 = 225
  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−3)(4x+3)=0\left(x - 3\right)\left(4x + 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−3=0⇒x=34x+3=0⇒x=−34x - 3 = 0 \Rightarrow x = 3\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±2258x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{8}
  7. 7

    Simplify to get the solutions:

    x=3orx=−34x = 3\quad\text{or}\quad x = -\frac{3}{4}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 4·(9) + (-9)·(3) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 9x - 9 = 0\ \Rightarrow\ x = 3,\ -\frac{3}{4} \]
\[ 4x^2 - 9x - 9 = 0 \]
\[ D = (-9)^2 - 4\cdot4\cdot(-9) = 81 + 144 = 225 \]
\[ \left(x - 3\right)\left(4x + 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{8} \]
\[ x = 3\quad\text{or}\quad x = -\frac{3}{4} \]

The parabola y = 4x² - 9x - 9

xy
-1.87521.9375
-0.8751.9375
0.125-10.0625
1.125-14.0625
2.125-10.0625
3.1251.9375
4.12521.9375

Questions

What are the solutions of 4x² - 9x - 9 = 0?

The solutions of 4x² - 9x - 9 = 0 are x = 3 or x = -3/4.

Does 4x² - 9x - 9 = 0 factor?

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

What is the vertex of y = 4x² - 9x - 9?

(9/8, -225/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9/4, product = c/a = -9/4.

Similar equations

    Solve 4x² - 9x - 9 = 0: x = 3 or x = -3/4 (steps) | WordGPT