WordGPT

Solve 4x² - 6x - 10 = 0

The solutions of 4x² - 6x - 10 = 0 are x = 5/2 or x = -1.

x² +x += 0
4x2−6x−10=0⇒x=52, −14x^2 - 6x - 10 = 0 \Rightarrow x = \frac{5}{2},\ -1
Discriminant D = 196 · x ≈ 2.5, -1

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 - 6x - 10-3-2-112345-20-10102030405060x = 2.5x = -1vertexy = 4x² - 6x - 10
Graph of y = 4x² - 6x - 10. The curve crosses the x-axis at x = 2.5 and x = -1; the vertex is at (0.75, -12.25).

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 = -6, c = -10.

    4x2−6x−10=04x^2 - 6x - 10 = 0
  2. 2

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

    D=(−6)2−4⋅4⋅(−10)=36+160=196D = (-6)^2 - 4\cdot4\cdot(-10) = 36 + 160 = 196
  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:

    2(2x−5)(x+1)=02\left(2x - 5\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:

    2x−5=0⇒x=52x+1=0⇒x=−12x - 5 = 0 \Rightarrow x = \frac{5}{2}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±1968x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{196}}{8}
  7. 7

    Simplify to get the solutions:

    x=52orx=−1x = \frac{5}{2}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 5/2 back in: a·x² + b·x + c = 4·(25/4) + (-6)·(5/2) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 6x - 10 = 0\ \Rightarrow\ x = \frac{5}{2},\ -1 \]
\[ 4x^2 - 6x - 10 = 0 \]
\[ D = (-6)^2 - 4\cdot4\cdot(-10) = 36 + 160 = 196 \]
\[ 2\left(2x - 5\right)\left(x + 1\right) = 0 \]
\[ 2x - 5 = 0 \Rightarrow x = \frac{5}{2}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{196}}{8} \]
\[ x = \frac{5}{2}\quad\text{or}\quad x = -1 \]

The parabola y = 4x² - 6x - 10

xy
-2.2523.75
-1.253.75
-0.25-8.25
0.75-12.25
1.75-8.25
2.753.75
3.7523.75

Questions

What are the solutions of 4x² - 6x - 10 = 0?

The solutions of 4x² - 6x - 10 = 0 are x = 5/2 or x = -1.

Does 4x² - 6x - 10 = 0 factor?

Yes: 2(2x-5)(x+1) = 0.

What is the vertex of y = 4x² - 6x - 10?

(3/4, -49/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = -5/2.

Similar equations

    Solve 4x² - 6x - 10 = 0: x = 5/2 or x = -1 (steps) | WordGPT