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Solve 4x² + 11x + 7 = 0

The solutions of 4x² + 11x + 7 = 0 are x = -1 or x = -7/4.

x² +x += 0
4x2+11x+7=0⇒x=−1, −744x^2 + 11x + 7 = 0 \Rightarrow x = -1,\ -\frac{7}{4}
Discriminant D = 9 · 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 + 11x + 7-4-3-2-115101520253035x = -1x = -1.75vertexy = 4x² + 11x + 7
Graph of y = 4x² + 11x + 7. The curve crosses the x-axis at x = -1 and x = -1.75; the vertex is at (-1.375, -0.5625).

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 = 7.

    4x2+11x+7=04x^2 + 11x + 7 = 0
  2. 2

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

    D=112−4⋅4⋅7=121−112=9D = 11^2 - 4\cdot4\cdot7 = 121 - 112 = 9
  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=−11±98x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{9}}{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) + 11·(-1) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 11x + 7 = 0\ \Rightarrow\ x = -1,\ -\frac{7}{4} \]
\[ 4x^2 + 11x + 7 = 0 \]
\[ D = 11^2 - 4\cdot4\cdot7 = 121 - 112 = 9 \]
\[ \left(x + 1\right)\left(4x + 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 4x + 7 = 0 \Rightarrow x = -\frac{7}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{9}}{8} \]
\[ x = -1\quad\text{or}\quad x = -\frac{7}{4} \]

The parabola y = 4x² + 11x + 7

xy
-4.37535.4375
-3.37515.4375
-2.3753.4375
-1.375-0.5625
-0.3753.4375
0.62515.4375
1.62535.4375

Questions

What are the solutions of 4x² + 11x + 7 = 0?

The solutions of 4x² + 11x + 7 = 0 are x = -1 or x = -7/4.

Does 4x² + 11x + 7 = 0 factor?

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

What is the vertex of y = 4x² + 11x + 7?

(-11/8, -9/16). 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 = 7/4.

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