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Solve 4x² - 7x + 3 = 0

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

x² +x += 0
4x2−7x+3=0⇒x=1, 344x^2 - 7x + 3 = 0 \Rightarrow x = 1,\ \frac{3}{4}
Discriminant D = 1 · x ≈ 1, 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 - 7x + 3-1123510152025x = 1x = 0.75vertexy = 4x² - 7x + 3
Graph of y = 4x² - 7x + 3. The curve crosses the x-axis at x = 1 and x = 0.75; the vertex is at (0.875, -0.0625).

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 = 3.

    4x2−7x+3=04x^2 - 7x + 3 = 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⋅3=49−48=1D = (-7)^2 - 4\cdot4\cdot3 = 49 - 48 = 1
  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−3)=0\left(x - 1\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−1=0⇒x=14x−3=0⇒x=34x - 1 = 0 \Rightarrow x = 1\qquad 4x - 3 = 0 \Rightarrow x = \frac{3}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±18x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{8}
  7. 7

    Simplify to get the solutions:

    x=1orx=34x = 1\quad\text{or}\quad x = \frac{3}{4}
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 - 7x + 3 = 0\ \Rightarrow\ x = 1,\ \frac{3}{4} \]
\[ 4x^2 - 7x + 3 = 0 \]
\[ D = (-7)^2 - 4\cdot4\cdot3 = 49 - 48 = 1 \]
\[ \left(x - 1\right)\left(4x - 3\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 4x - 3 = 0 \Rightarrow x = \frac{3}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{8} \]
\[ x = 1\quad\text{or}\quad x = \frac{3}{4} \]

The parabola y = 4x² - 7x + 3

xy
-2.12535.9375
-1.12515.9375
-0.1253.9375
0.875-0.0625
1.8753.9375
2.87515.9375
3.87535.9375

Questions

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

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

Does 4x² - 7x + 3 = 0 factor?

Yes: (x-1)(4x-3) = 0.

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

(7/8, -1/16). 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 = 3/4.

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