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Solve 4x² + 3x - 7 = 0

The solutions of 4x² + 3x - 7 = 0 are x = 1 or x = -7/4.

x² +x += 0
4x2+3x−7=0⇒x=1, −744x^2 + 3x - 7 = 0 \Rightarrow x = 1,\ -\frac{7}{4}
Discriminant D = 121 · x ≈ 1, -1.75

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

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Graph of y = 4x² + 3x - 7-4-3-2-1123-101020304050x = 1x = -1.75vertexy = 4x² + 3x - 7
Graph of y = 4x² + 3x - 7. The curve crosses the x-axis at x = 1 and x = -1.75; the vertex is at (-0.375, -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 = 3, c = -7.

    4x2+3x−7=04x^2 + 3x - 7 = 0
  2. 2

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

    D=32−4⋅4⋅(−7)=9+112=121D = 3^2 - 4\cdot4\cdot(-7) = 9 + 112 = 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−1)(4x+7)=0\left(x - 1\right)\left(4x + 7\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−1=0⇒x=14x+7=0⇒x=−74x - 1 = 0 \Rightarrow x = 1\qquad 4x + 7 = 0 \Rightarrow x = -\frac{7}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±1218x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{121}}{8}
  7. 7

    Simplify to get the solutions:

    x=1orx=−74x = 1\quad\text{or}\quad x = -\frac{7}{4}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 4·(1) + 3·(1) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 3x - 7 = 0\ \Rightarrow\ x = 1,\ -\frac{7}{4} \]
\[ 4x^2 + 3x - 7 = 0 \]
\[ D = 3^2 - 4\cdot4\cdot(-7) = 9 + 112 = 121 \]
\[ \left(x - 1\right)\left(4x + 7\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 4x + 7 = 0 \Rightarrow x = -\frac{7}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{121}}{8} \]
\[ x = 1\quad\text{or}\quad x = -\frac{7}{4} \]

The parabola y = 4x² + 3x - 7

xy
-3.37528.4375
-2.3758.4375
-1.375-3.5625
-0.375-7.5625
0.625-3.5625
1.6258.4375
2.62528.4375

Questions

What are the solutions of 4x² + 3x - 7 = 0?

The solutions of 4x² + 3x - 7 = 0 are x = 1 or x = -7/4.

Does 4x² + 3x - 7 = 0 factor?

Yes: (x-1)(4x+7) = 0.

What is the vertex of y = 4x² + 3x - 7?

(-3/8, -121/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3/4, product = c/a = -7/4.

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