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Solve 4x² - 7x - 2 = 0

The solutions of 4x² - 7x - 2 = 0 are x = 2 or x = -1/4.

x² +x += 0
4x2−7x−2=0⇒x=2, −144x^2 - 7x - 2 = 0 \Rightarrow x = 2,\ -\frac{1}{4}
Discriminant D = 81 · x ≈ 2, -0.25

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

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Graph of y = 4x² - 7x - 2-2-11234-101020304050x = 2x = -0.25vertexy = 4x² - 7x - 2
Graph of y = 4x² - 7x - 2. The curve crosses the x-axis at x = 2 and x = -0.25; the vertex is at (0.875, -5.0625).

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 = -7, c = -2.

    4x2−7x−2=04x^2 - 7x - 2 = 0
  2. 2

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

    D=(−7)2−4⋅4⋅(−2)=49+32=81D = (-7)^2 - 4\cdot4\cdot(-2) = 49 + 32 = 81
  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+1)=0\left(x - 2\right)\left(4x + 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:

    x−2=0⇒x=24x+1=0⇒x=−14x - 2 = 0 \Rightarrow x = 2\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±818x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{81}}{8}
  7. 7

    Simplify to get the solutions:

    x=2orx=−14x = 2\quad\text{or}\quad x = -\frac{1}{4}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 4·(4) + (-7)·(2) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 7x - 2 = 0\ \Rightarrow\ x = 2,\ -\frac{1}{4} \]
\[ 4x^2 - 7x - 2 = 0 \]
\[ D = (-7)^2 - 4\cdot4\cdot(-2) = 49 + 32 = 81 \]
\[ \left(x - 2\right)\left(4x + 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{81}}{8} \]
\[ x = 2\quad\text{or}\quad x = -\frac{1}{4} \]

The parabola y = 4x² - 7x - 2

xy
-2.12530.9375
-1.12510.9375
-0.125-1.0625
0.875-5.0625
1.875-1.0625
2.87510.9375
3.87530.9375

Questions

What are the solutions of 4x² - 7x - 2 = 0?

The solutions of 4x² - 7x - 2 = 0 are x = 2 or x = -1/4.

Does 4x² - 7x - 2 = 0 factor?

Yes: (x-2)(4x+1) = 0.

What is the vertex of y = 4x² - 7x - 2?

(7/8, -81/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/4, product = c/a = -1/2.

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    Solve 4x² - 7x - 2 = 0: x = 2 or x = -1/4 (steps) | WordGPT