Problem 17 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ x^{2}+18=10 x $$

Step-by-Step Solution

Answer

The solutions are \( x = 5 + \sqrt{7} \) and \( x = 5 - \sqrt{7} \).

Step 1 - Rewrite the equation in standard form

Move all terms to one side of the equation to set it to zero: \[ x^{2} - 10x + 18 = 0 \]

Step 2 - Identify coefficients

Identify the coefficients from the standard form equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -10 \), and \( c = 18 \).

Step 3 - Write the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 4 - Substitute the coefficients into the quadratic formula

Substitute \( a = 1 \), \( b = -10 \), and \( c = 18 \) into the quadratic formula: \[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(18)}}{2(1)} \]

Step 5 - Simplify under the square root

Calculate the discriminant \( b^2 - 4ac \): \[ (-10)^2 - 4(1)(18) = 100 - 72 = 28 \] Therefore, the equation becomes: \[ x = \frac{10 \pm \sqrt{28}}{2} \]

Step 6 - Simplify the square root and final expression

Simplify \( \sqrt{28} = 2\sqrt{7} \), then the equation is: \[ x = \frac{10 \pm 2\sqrt{7}}{2} \] Divide the terms by 2: \[ x = 5 \pm \sqrt{7} \]

Step 7 - Write the final solutions

The solutions to the equation \( x^{2} - 10x + 18 = 0 \) are: \[ x = 5 + \sqrt{7} \] and \[ x = 5 - \sqrt{7} \]

Key Concepts

solving quadratic equations

Solving quadratic equations might seem tricky, but it's manageable if you break it down step by step. First, ensure your equation is in standard form, which looks like this: \[ ax^2 + bx + c = 0 \]. The terms must be on one side, and the other side should be zero. This form makes it easier to apply the quadratic formula.
Let's say our equation is\( x^2 + 18 = 10x \). Move all terms so one side equals zero: \[ x^2 -10x + 18 = 0. \] Now you're ready for the next steps!

discriminant

The discriminant is a crucial part of the quadratic formula. It determines the nature of the solutions you'll get. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. The term under the square root, \[ b^2 - 4ac \], is the discriminant.
Depending on its value, it tells us:

• If \[ b^2 - 4ac > 0 \]: two real and distinct solutions.
• If \[ b^2 - 4ac = 0 \]: one real and repeated solution.
• If \[ b^2 - 4ac < 0 \]: no real solutions (only complex ones).

For the equation \[ x^2 - 10x + 18 = 0 \], the discriminant calculation goes like this: \[ (-10)^2 - 4 \times 1 \times 18 = 100 - 72 = 28 \]. Since 28>0, we get two distinct real solutions.

simplifying radicals

Simplifying radicals is often necessary when solving quadratic equations. When you have a square root, you may need to simplify it. For example, in our quadratic formula, after substituting the coefficients, we get \[ \frac{10 \pm \sqrt{28}}{2} \]. The term \[ \sqrt{28} \] can be simplified since \[ 28 = 4 \times 7 \]. So, it becomes \[ \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7} \].
Substitute this back into the quadratic formula: \[ x = \frac{10 \pm 2\sqrt{7}}{2} \]. Now, divide all terms by 2: \[ x = 5 \pm \sqrt{7} \]. Thus, the solutions are: \[ x = 5 + \sqrt{7} \] and \[ x = 5 - \sqrt{7} \].
Simplifying radicals helps in getting the final, clean form of the answer.