Q. Complete the square to rewrite the quadratic function in vertex form: \(y = 8x^2 – 48x + 69\).

Q: What is the vertex form of a quadratic?

Vertex form is y=a(x-h)^2+k, where (h,k) is the vertex and a controls width and direction.

Q: How do I start completing the square for y=8x^2-48x+69?

Factor out 8 from the quadratic terms: y=8(x^2-6x)+69..

Q: How do I complete the square for x^2-6x?

Take half of -6 (which is -3 ), square it to get 9 . Write x^2-6x=(x-3)^2-9 .

Q: What is the vertex form after completing the square?

Substituting gives y=8\big((x-3)^2-9\big)+69, which simplifies to y=8(x-3)^2-3.

Q: What is the vertex of the parabola?

The vertex is (3,-3) from y=8(x-3)^2-3 .

Q: Does the parabola open up or down and what's the extremum?

Since a=8>0, it opens up and has a minimum value y=-3 at x=3.

Q: How can I check my work?

Expand 8(x-3)^2-3 to get 8x^2-48x+69. If you retrieve the original, the conversion is correct.

Answer

y = 8(x^2 − 6x) + 69
y = 8[(x − 3)^2 − 9] + 69
y = 8(x − 3)^2 − 3

Final result: y = 8(x − 3)^2 − 3

Detailed Explanation

Complete the square to write the quadratic in vertex form

Start with the given quadratic function
y = 8x^2 - 48x + 69
Factor out the leading coefficient 8 from the x‑terms
y = 8(x^2 - 6x) + 69
To complete the square inside the parentheses, take half of the coefficient of x (which is -6), and square it:
Half of -6 is -3, and (-3)^2 = 9.
Add and subtract that square inside the parentheses (this does not change the expression):
y = 8(x^2 - 6x + 9 - 9) + 69
Group the perfect square trinomial and the remaining constant inside the brackets:
y = 8[(x^2 - 6x + 9) - 9] + 69

The trinomial x^2 – 6x + 9 is (x – 3)^2, so

y = 8[(x - 3)^2 - 9] + 69
Distribute the 8 across the bracketed terms:
y = 8(x - 3)^2 - 8·9 + 69

y = 8(x - 3)^2 - 72 + 69
Simplify the constants:
y = 8(x - 3)^2 - 3
Therefore the vertex form is
y = 8(x - 3)^2 - 3

The vertex is (3, -3). The parabola opens upward (since 8 > 0) and is narrower than y = x^2 because the leading coefficient 8 > 1.

See full solution

Master vertex form with AI, try our homework help tools now!

Homework helper

Algebra FAQs

What is the vertex form of a quadratic?

Vertex form is \(y=a(x-h)^2+k\), where \((h,k)\) is the vertex and \(a\) controls width and direction.

How do I start completing the square for \(y=8x^2-48x+69\)?

Factor out 8 from the quadratic terms: \(y=8(x^2-6x)+69\)..

How do I complete the square for \(x^2-6x\)?

Take half of \( -6 \) (which is \( -3 \)), square it to get \( 9 \). Write \( x^2-6x=(x-3)^2-9 \).

What is the vertex form after completing the square?

Substituting gives \(y=8\big((x-3)^2-9\big)+69\), which simplifies to \(y=8(x-3)^2-3\).

What is the vertex of the parabola?

The vertex is \( (3,-3) \) from \( y=8(x-3)^2-3 \).

What is the axis of symmetry?

Does the parabola open up or down and what's the extremum?

Since \(a=8>0\), it opens up and has a minimum value \(y=-3\) at \(x=3\).

How can I check my work?

Expand \(8(x-3)^2-3\) to get \(8x^2-48x+69\). If you retrieve the original, the conversion is correct.

Use our AI tools for homework help.
Finance, econ, and accounting tools.

252,312+ customers tried
Analytical, General, Biochemistry, etc.

Finance homework help Economics homework help Accounting homework help