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Solve 4x² + 7x + 3 = 0

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

x² +x += 0
4x2+7x+3=0⇒x=−34, −14x^2 + 7x + 3 = 0 \Rightarrow x = -\frac{3}{4},\ -1
Discriminant D = 1 · x ≈ -0.75, -1

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

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Graph of y = 4x² + 7x + 3-3-2-11510152025x = -0.75x = -1vertexy = 4x² + 7x + 3
Graph of y = 4x² + 7x + 3. The curve crosses the x-axis at x = -0.75 and x = -1; 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=72−4⋅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:

    (4x+3)(x+1)=0\left(4x + 3\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+3=0⇒x=−34x+1=0⇒x=−14x + 3 = 0 \Rightarrow x = -\frac{3}{4}\qquad x + 1 = 0 \Rightarrow x = -1
  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=−34orx=−1x = -\frac{3}{4}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -3/4 back in: a·x² + b·x + c = 4·(9/16) + 7·(-3/4) + 3 = 0 ✓

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

The parabola y = 4x² + 7x + 3

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

Questions

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

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

Does 4x² + 7x + 3 = 0 factor?

Yes: (4x+3)(x+1) = 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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