Question

y = 2x² - 20x + 18

Explanation:

Assuming we want to find the vertex of the quadratic function \( y = 2x^2 - 20x + 18 \) (a common problem with quadratic equations), here's the step - by - step solution:

Step 1: Recall the formula for the x - coordinate of the vertex of a quadratic function \( y = ax^2+bx + c \)

The formula for the x - coordinate of the vertex is \( x=-\frac{b}{2a} \). For the function \( y = 2x^2-20x + 18 \), we have \( a = 2 \) and \( b=- 20 \).
\( x=-\frac{-20}{2\times2}=\frac{20}{4} = 5 \)

Step 2: Find the y - coordinate of the vertex by substituting \( x = 5 \) into the function

Substitute \( x = 5 \) into \( y=2x^2-20x + 18 \):
\( y=2\times(5)^2-20\times(5)+18 \)
\( y=2\times25-100 + 18 \)
\( y = 50-100 + 18=-32 \)

If we want to factor the quadratic:

Step 1: Factor out the common factor from the first two terms

Given \( y = 2x^2-20x + 18 \), factor out 2 from the first two terms: \( y=2(x^2 - 10x)+18 \)

Step 2: Complete the square inside the parentheses

For the expression \( x^2-10x \), we know that \( (x - h)^2=x^2-2hx+h^2 \). If \( - 2h=-10 \), then \( h = 5 \) and \( h^2 = 25 \). So we add and subtract 25 inside the parentheses:
\( y=2(x^2-10x + 25-25)+18 \)
\( y=2((x - 5)^2-25)+18 \)

Step 3: Distribute the 2 and simplify

\( y=2(x - 5)^2-50 + 18 \)
\( y=2(x - 5)^2-32 \)

If we want to find the roots (solve \( 2x^2-20x + 18 = 0 \)):

Step 1: Divide the entire equation by 2

We get \( x^2-10x + 9=0 \)

Step 2: Factor the quadratic equation

We need two numbers that multiply to 9 and add up to - 10. The numbers are - 1 and - 9. So \( x^2-10x + 9=(x - 1)(x - 9)=0 \)

Step 3: Solve for x

Set each factor equal to zero:
\( x - 1=0\) or \(x - 9=0\)
So \( x = 1 \) or \( x=9 \)

Since the problem was not specified, if we assume finding the vertex, the vertex of the parabola \( y = 2x^2-20x + 18 \) is at the point \( (5,-32) \). If we assume factoring, the factored form is \( y = 2(x - 1)(x - 9) \) (or \( y=2(x - 5)^2-32 \) in vertex form). If we assume finding the roots, the roots are \( x = 1 \) and \( x = 9 \)

If you want to perform a different operation (like finding the axis of symmetry, determining if it opens up or down etc.), here are the relevant steps:

• Axis of symmetry: For a quadratic \( y = ax^2+bx + c \), the axis of symmetry is \( x=-\frac{b}{2a} \). Using \( a = 2 \), \( b=-20 \), we have \( x = 5 \) (same as the x - coordinate of the vertex).
• Direction of opening: Since \( a=2>0 \), the parabola opens upwards.

If we take the vertex - finding case as the main problem:

Answer:

The vertex of the parabola \( y = 2x^2-20x + 18 \) is \( (5,-32) \)