Converting Quadratics: Standard Form and Vertex Form
Completing the square turns standard form, y = ax² + bx + c, into vertex form, y = a(x − h)² + k: two ways of writing the same quadratic. Expanding the brackets changes it back.
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What are Standard Form and Vertex Form?
- Standard (general) form:
- Vertex form:
Converting Vertex Form to Standard Form
- Expand the square using , then expand the brackets
- Don't forget the constant at the end (this is a very common mistake)
Converting Standard Form to Vertex Form
- Only factor out from the and terms. Ignore the constant
- E.g.
Completing the Square
- When completing the square, anything added inside must be undone outside
- E.g.
Practice Questions
Test your understanding
Convert the quadratic equation into general form.
Correct! 🎉 +10 pointsCorrect! 🎉 Worth 10 XPNot quite right
To convert from vertex form to general form, start by expanding to get . Then distribute the 3 to every term: .
Finally, subtract the 4 that sits outside the brackets: .
Simplifying, we get .
Convert the quadratic equation into general form.
Correct! 🎉 +10 pointsCorrect! 🎉 Worth 10 XPNot quite right
To convert from vertex form to general form, start by expanding to get . Then distribute the 4 to every term: .
Finally, add the 2 that sits outside the brackets: .
Simplifying, we get .
Convert the quadratic equation into vertex form.
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Complete the square on the terms, carrying the through every line.
Half of 4 is 2, and , so . The cancels the extra 4 that squaring the bracket creates.
Put that back into the equation, keeping the : .
Simplifying, .
Convert the quadratic equation into vertex form.
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Complete the square on the terms, keeping the throughout. Half of is , and , so .
Put that back into the equation: .
Simplifying, .
Convert the quadratic equation into vertex form.
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Complete the square on the terms, keeping the throughout. Half of is , and , so .
Put that back into the equation: .
Simplifying, .
Convert the quadratic equation into vertex form.
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First factor 3 out of the and terms: .
Complete the square inside the brackets. Half of is , and , so . That gives .
Multiplying the 3 back into everything inside the brackets, the becomes to subtract: .
Simplifying, .
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Interactive Activity
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Frequently asked questions
The questions students bump into most on this topic
Completing the square rewrites a quadratic from the general form, ax² + bx + c, into the vertex form, a(x − h)² + k, by turning the x² and x terms into a squared bracket. It shows the turning point straight away, which helps with sketching and with solving. It is a Higher-tier GCSE topic, usually placed at grades 7 to 9.
Halve the number in front of x, write a squared bracket with that half, then subtract its square. For x² + 4x + 5: half of 4 is 2, so write (x + 2)², which is x² + 4x + 4; subtract the extra 4 and keep the + 5, giving (x + 2)² + 1. To check, expand your answer and see that you get back the original expression.
Factor the number in front of x² out of the x² and x terms only, then complete the square inside the bracket. For 3x² + 18x + 25: write 3(x² + 6x) + 25, then half of 6 is 3, so 3[(x + 3)² − 9] + 25 = 3(x + 3)² − 27 + 25 = 3(x + 3)² − 2. The same works when a is negative: keep the sign when you factor it out.
Read it off the vertex form: y = a(x − h)² + k has its turning point at (h, k). For y = 3(x + 3)² − 2, the bracket is zero when x = −3, so the turning point is (−3, −2), a minimum because a is positive. For y = −2(x − 2)² + 3 it is (2, 3), a maximum because a is negative. Watch the sign: (x + 3) means h = −3.
Expand the squared bracket, multiply by the number outside, then add the constant and collect like terms. For y = 2(x − 3)² − 5: (x − 3)² = x² − 6x + 9, so 2(x² − 6x + 9) − 5 = 2x² − 12x + 18 − 5 = 2x² − 12x + 13. Forgetting the constant at the end is the common slip. American sites call the general form standard form.
Because squaring the bracket adds a number that was not there. (x + 3)² is x² + 6x + 9, but the original expression only had x² + 6x, so you subtract the extra 9 to keep the value the same. Halving the coefficient of x is what makes the bracket produce the right x term: 2 × 3 = 6.
What's next?
Found this one hard? Go one step back first. Feeling confident? Carry on to the next lesson.
