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Chapter 33: Problem 972

Use the quadratic formula to solve for \(\mathrm{x}\) in the equation \(x^{2}-5 x+6=0\)

Short Answer

Expert verified
The solutions for the given quadratic equation \(x^2-5x+6=0\) are \(x_1 = 3\) and \(x_2 = 2\), obtained using the quadratic formula with \(a=1\), \(b=-5\), and \(c=6\).

Step by step solution

01

Identify the coefficients a, b, and c

From the given quadratic equation \(x^2-5x+6=0\), we can see the coefficients are: \(a = 1\), \(b = -5\), and \(c = 6\).
02

Apply the quadratic formula

Now that we know our coefficients, let's apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Substitute the coefficients \(a\), \(b\), and \(c\) into the formula: \(x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}\)
03

Simplify the expression

Simplify the expression within the formula: \(x = \frac{5 \pm \sqrt{25 - 24}}{2}\) \(x = \frac{5 \pm \sqrt{1}}{2}\)
04

Calculate the two values of x

As we have a 卤 symbol, we have two different solutions for x. Calculate the two values of x using the positive and negative square roots: For the positive square root: \(x = \frac{5 + \sqrt{1}}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3\) For the negative square root: \(x = \frac{5 - \sqrt{1}}{2} = \frac{5 - 1}{2} = \frac{4}{2} = 2\)
05

Write down the solutions for x

The two solutions for the given quadratic equation are: \(x_1 = 3\) and \(x_2 = 2\).

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