WordGPT

Solve 4x² + 11x - 3 = 0

The solutions of 4x² + 11x - 3 = 0 are x = 1/4 or x = -3.

x² +x += 0
4x2+11x−3=0⇒x=14, −34x^2 + 11x - 3 = 0 \Rightarrow x = \frac{1}{4},\ -3
Discriminant D = 169 · x ≈ 0.25, -3

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 + 11x - 3-5-4-3-2-112-10102030405060x = 0.25x = -3vertexy = 4x² + 11x - 3
Graph of y = 4x² + 11x - 3. The curve crosses the x-axis at x = 0.25 and x = -3; the vertex is at (-1.375, -10.5625).

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 = 11, c = -3.

    4x2+11x−3=04x^2 + 11x - 3 = 0
  2. 2

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

    D=112−4⋅4⋅(−3)=121+48=169D = 11^2 - 4\cdot4\cdot(-3) = 121 + 48 = 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:

    (4x−1)(x+3)=0\left(4x - 1\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:

    4x−1=0⇒x=14x+3=0⇒x=−34x - 1 = 0 \Rightarrow x = \frac{1}{4}\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1698x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{169}}{8}
  7. 7

    Simplify to get the solutions:

    x=14orx=−3x = \frac{1}{4}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 1/4 back in: a·x² + b·x + c = 4·(1/16) + 11·(1/4) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 11x - 3 = 0\ \Rightarrow\ x = \frac{1}{4},\ -3 \]
\[ 4x^2 + 11x - 3 = 0 \]
\[ D = 11^2 - 4\cdot4\cdot(-3) = 121 + 48 = 169 \]
\[ \left(4x - 1\right)\left(x + 3\right) = 0 \]
\[ 4x - 1 = 0 \Rightarrow x = \frac{1}{4}\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{169}}{8} \]
\[ x = \frac{1}{4}\quad\text{or}\quad x = -3 \]

The parabola y = 4x² + 11x - 3

xy
-4.37525.4375
-3.3755.4375
-2.375-6.5625
-1.375-10.5625
-0.375-6.5625
0.6255.4375
1.62525.4375

Questions

What are the solutions of 4x² + 11x - 3 = 0?

The solutions of 4x² + 11x - 3 = 0 are x = 1/4 or x = -3.

Does 4x² + 11x - 3 = 0 factor?

Yes: (4x-1)(x+3) = 0.

What is the vertex of y = 4x² + 11x - 3?

(-11/8, -169/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/4, product = c/a = -3/4.

Similar equations

    Solve 4x² + 11x - 3 = 0: x = 1/4 or x = -3 (steps) | WordGPT