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Solve 4x² - 11x + 7 = 0

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

x² +x += 0
4x2−11x+7=0⇒x=74, 14x^2 - 11x + 7 = 0 \Rightarrow x = \frac{7}{4},\ 1
Discriminant D = 9 · x ≈ 1.75, 1

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

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Graph of y = 4x² - 11x + 7-112345101520253035x = 1.75x = 1vertexy = 4x² - 11x + 7
Graph of y = 4x² - 11x + 7. The curve crosses the x-axis at x = 1.75 and x = 1; 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=(−11)2−4⋅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:

    (4x−7)(x−1)=0\left(4x - 7\right)\left(x - 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:

    4x−7=0⇒x=74x−1=0⇒x=14x - 7 = 0 \Rightarrow x = \frac{7}{4}\qquad x - 1 = 0 \Rightarrow x = 1
  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=74orx=1x = \frac{7}{4}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = 7/4 back in: a·x² + b·x + c = 4·(49/16) + (-11)·(7/4) + 7 = 0 ✓

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

The parabola y = 4x² - 11x + 7

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

Questions

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

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

Does 4x² - 11x + 7 = 0 factor?

Yes: (4x-7)(x-1) = 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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