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Solve the following quadratic equations by using the formula, giving the solutions in surd form. Simplify your answers.

Mathematics

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Answer

x=−3±52x = \frac{-3 \pm \sqrt{5}}{2}

Step-by-step solution

3 steps

Here's how to solve the first quadratic equation using the formula:

The quadratic equation is x2+3x+1=0x^2 + 3x + 1 = 0.
The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0.
Comparing the given equation with the general form, we have:
a=1a = 1
b=3b = 3
c=1c = 1

The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  1. Substitute the values of aa, bb, and cc into the quadratic formula.

    x=−(3)±(3)2−4(1)(1)2(1)x = \frac{-(3) \pm \sqrt{(3)^2 - 4(1)(1)}}{2(1)}

  2. Simplify the expression under the square root (the discriminant).

    x=−3±9−42x = \frac{-3 \pm \sqrt{9 - 4}}{2} x=−3±52x = \frac{-3 \pm \sqrt{5}}{2}

  3. Write out the two solutions.

    The solutions are:

    x1=−3+52x_1 = \frac{-3 + \sqrt{5}}{2} x2=−3−52x_2 = \frac{-3 - \sqrt{5}}{2}
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