Solve x² + 7x + 3 = 0
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = (−7 − √37)/2, x = (√37 − 7)/2 |
Step-by-step solution
8 steps-
1 Givenx^{2} + 7 x + 3 = 0
Solve for x.
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2 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = 7 and c = 3, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 7,\quad c = 3 -
3 Apply the quadratic formula
The quadratic formula solves any equation ax² + bx + c = 0. It comes from completing the square on the general equation, so it always works, even when the trinomial does not factor.
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} -
4 Substitute a, b and cx = \frac{-7 \pm \sqrt{7^{2} - 4 \cdot 1 \cdot 3}}{2 \cdot 1}
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5 Compute the discriminant D = b² − 4ac
D = b² − 4ac is the part under the square root in the formula. If it is positive there are two real solutions, if it is 0 there is one, and if it is negative its square root is not a real number, so there is no real solution.
D = 37Positive discriminant: two distinct real solutions.
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6 Evaluate the square root and the denominatorx = \frac{-7 \pm \sqrt{37}}{2}
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7 Simplifyx = \frac{-7 - \sqrt{37}}{2}\quad \text{or} \quad x = \frac{\sqrt{37} - 7}{2}
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8 Check\begin{aligned}x = \frac{-7 - \sqrt{37}}{2}:\quad 0 = 0\quad\checkmark\\ x = \frac{\sqrt{37} - 7}{2}:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| (−7 − √37)/2 | 0 | 0 | ✓ |
| (√37 − 7)/2 | 0 | 0 | ✓ |
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