Question

solve by using the quadratic formula.

\\2x^2 - 18x + 45 = 5\\

\\x = \text{ } , \text{ }\\

Explanation:

🆕 New Concept Discovered: Quadratic Formula
Using a universal formula to solve any quadratic equation

Step 1: Standard Form

To use the quadratic formula, we must first set the equation to equal zero:

$$ 2x^2 - 18x + 45 = 5 $$

Subtract \( 5 \) from both sides:

$$ 2x^2 - 18x + 40 = 0 $$

Step 2: Identify Coefficients

Identify the values of \( a \), \( b \), and \( c \) from the standard form \( ax^2 + bx + c = 0 \):

$$ a = 2, \quad b = -18, \quad c = 40 $$

Step 3: Apply the Quadratic Formula

The quadratic formula is a reliable tool that always finds the solutions for any quadratic equation:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Substitute the values into the formula:

$$ x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(2)(40)}}{2(2)} $$

$$ x = \frac{18 \pm \sqrt{324 - 320}}{4} $$

$$ x = \frac{18 \pm \sqrt{4}}{4} $$

Step 4: Simplify and Solve

Simplify the square root and calculate the two possible values for \( x \):

$$ x = \frac{18 \pm 2}{4} $$

Case 1 (addition):

$$ x = \frac{18 + 2}{4} = \frac{20}{4} = 5 $$

Case 2 (subtraction):

$$ x = \frac{18 - 2}{4} = \frac{16}{4} = 4 $$

Answer:

$$ x = 4, 5 $$