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Solve 4x² - 5x - 6 = 0

The solutions of 4x² - 5x - 6 = 0 are x = 2 or x = -3/4.

x² +x += 0
4x2−5x−6=0⇒x=2, −344x^2 - 5x - 6 = 0 \Rightarrow x = 2,\ -\frac{3}{4}
Discriminant D = 121 · x ≈ 2, -0.75

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

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Graph of y = 4x² - 5x - 6-3-2-11234-101020304050x = 2x = -0.75vertexy = 4x² - 5x - 6
Graph of y = 4x² - 5x - 6. The curve crosses the x-axis at x = 2 and x = -0.75; the vertex is at (0.625, -7.5625).

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 = -5, c = -6.

    4x2−5x−6=04x^2 - 5x - 6 = 0
  2. 2

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

    D=(−5)2−4⋅4⋅(−6)=25+96=121D = (-5)^2 - 4\cdot4\cdot(-6) = 25 + 96 = 121
  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−2)(4x+3)=0\left(x - 2\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−2=0⇒x=24x+3=0⇒x=−34x - 2 = 0 \Rightarrow x = 2\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1218x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{8}
  7. 7

    Simplify to get the solutions:

    x=2orx=−34x = 2\quad\text{or}\quad x = -\frac{3}{4}
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 - 5x - 6 = 0\ \Rightarrow\ x = 2,\ -\frac{3}{4} \]
\[ 4x^2 - 5x - 6 = 0 \]
\[ D = (-5)^2 - 4\cdot4\cdot(-6) = 25 + 96 = 121 \]
\[ \left(x - 2\right)\left(4x + 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{8} \]
\[ x = 2\quad\text{or}\quad x = -\frac{3}{4} \]

The parabola y = 4x² - 5x - 6

xy
-2.37528.4375
-1.3758.4375
-0.375-3.5625
0.625-7.5625
1.625-3.5625
2.6258.4375
3.62528.4375

Questions

What are the solutions of 4x² - 5x - 6 = 0?

The solutions of 4x² - 5x - 6 = 0 are x = 2 or x = -3/4.

Does 4x² - 5x - 6 = 0 factor?

Yes: (x-2)(4x+3) = 0.

What is the vertex of y = 4x² - 5x - 6?

(5/8, -121/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/4, product = c/a = -3/2.

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    Solve 4x² - 5x - 6 = 0: x = 2 or x = -3/4 (steps) | WordGPT