Hard GCSE Turning points by completing square Questions

Identify the coefficients

Compare $-3x^2 + 12x - 7$ with $ax^2 + bx + c$.

Take the factor of $-3$ out of the x-terms

Only the $x^2$ and x terms go inside the bracket; 12 $\div -3 = -4$.

Halve the coefficient of x

Half of $-4$ is $-2$. This is the number that goes inside the bracket.

Expand the squared bracket to see what it gives

Squaring the bracket produces the $x^2$ and $-4x$ terms, but it also adds $4$.

Rearrange to get the x-terms on their own

Subtract the unwanted $4$ from both sides.

Substitute back inside the bracket

The bracket that was multiplied by $-3$ is now written as a completed square.

Multiply the $-3$ through the outer bracket

$-3$ $\times -4 = 12$.

Write the completed square

$12 - 7 = 5$, giving $-3(x - 2)^2 + 5$.

Decide on the shape of the curve

The coefficient of $x^2$ is negative, so the parabola is n-shaped and the turning point is a maximum.

Use the fact that a square is never negative

Multiplying by the negative number $-3$ makes the term at most $0$, and it equals $0$ when $x = 2$.

Find the value of x at the turning point

The bracket is zero when $x = 2$.

Find the greatest value of y

With the squared term equal to 0, y takes its greatest value 5.

Check by substituting back into the original

Substituting $x = 2$ into the original equation gives $y = 5$, which confirms the turning point.

State the line of symmetry

A parabola is symmetrical about the vertical line through its turning point, so the line of symmetry is $x = 2$.

State the maximum point

The maximum point of the curve is ($2$,\ $5$).

Identify the coefficients

Compare $4x^2 + 12x + 5$ with $ax^2 + bx + c$.

Take the factor of $4$ out of the x-terms

Only the $x^2$ and x terms go inside the bracket; $12 \div 4 = 3$.

Halve the coefficient of x

Half of $3$ is $3/2$. This is the number that goes inside the bracket.

Expand the squared bracket to see what it gives

Squaring the bracket produces the $x^2$ and $3x$ terms, but it also adds $\frac{9}{4}$.

Rearrange to get the x-terms on their own

Subtract the unwanted $\frac{9}{4}$ from both sides.

Substitute back inside the bracket

The bracket that was multiplied by $4$ is now written as a completed square.

Multiply the $4$ through the outer bracket

$4$ $\times -\frac{9}{4} = -9$.

Write the completed square

$-9 + 5 = -4$, giving $4(x + \frac{3}{2})^2 - 4$.

Set the completed square equal to zero

The equation $4x^2 + 12x + 5 = 0$ becomes this.

Add $4$ to both sides

Move the constant across.

Divide both sides by $4$

Isolate the squared bracket.

Take the square root of both sides

Remember both square roots of $1$.

Solve for x

Subtract $3/2$ from each of +$1$ and $-1$.

Check by factorising

Expanding ($2x + 1$)($2x + 5$) gives $4x^2 + 10x + 2x + 5 = 4x^2 + 12x + 5$, and the roots agree.

Write down the smaller solution

Since $-5/2 = -2.5$ is less than $-1/2 = -0.5$, the smaller solution is $-5/2$.

Write an expression for the other side

The perimeter is $24$ cm, so two adjacent sides add up to $12$ cm.

Write an expression for the area

Area of a rectangle = length x width.

Expand the bracket

Multiply x by each term inside the bracket.

Write in the usual order

The coefficient of $x^2$ is negative, so A has a maximum.

Take out the factor of $-1$

Every term inside the bracket changes sign.

Halve the coefficient of x

Half of -12 is -6, so the bracket will be (x - 6).

Expand the squared bracket

Squaring adds an unwanted $36$.

Rearrange

Subtract the extra $36$.

Substitute back

Put the completed square inside the outer bracket.

Multiply out the minus sign

Both terms inside the bracket change sign.

Use the fact that a square is never negative

So A is never more than $36$.

Find the value of x giving the maximum

The area is greatest when $x = 6$ cm.

Find the other side

The other side is also $6$ cm, so the rectangle is a square.

Check the area

A $6$ cm by $6$ cm square has area $36\text{ cm}^2$ and perimeter $24$ cm.

State the greatest area

The area is greatest, at $36\text{ cm}^2$, when the rectangle is a $6$ cm square.

Identify the coefficients

Compare $x^2 - 10x + 29$ with $ax^2 + bx + c$.

Halve the coefficient of x

Half of $-10$ is $-5$. This is the number that goes inside the bracket.

Expand the squared bracket to see what it gives

Squaring the bracket produces the $x^2$ and $-10x$ terms, but it also adds $25$.

Rearrange to get the x-terms on their own

Subtract the unwanted $25$ from both sides.

Put the constant term back in

Replace $x^2 - 10x$ by $(x - 5)^2 - 25$ and keep the + $29$.

Simplify the constants

$-25 + 29 = 4$, so the completed square is $(x - 5)^2 + 4$.

Use the fact that a square is never negative

For every real value of x the square is 0 or more.

Add $4$ to both sides

Adding $4$ keeps the inequality true.

Interpret the inequality

Every point on the curve has y-coordinate at least $4$.

Find the turning point

The minimum point of the curve is ($5$, $4$).

Compare with the x-axis

A point on the x-axis has y-coordinate $0$.

Show $y = 0$ is impossible

The smallest y-value on the curve is $4$, which is greater than $0$.

Check with the discriminant

A negative discriminant confirms there are no real roots.

Interpret the discriminant

The curve never meets the x-axis.

Write the conclusion

Since y is always at least 4, the curve lies entirely above the x-axis, as required.

Write down the turning point

Whatever the value of a, the turning point of $a(x - 3)^2 + 5$ is at ($3$, $5$).

Use the point on the curve

Substitute $x = 1$ and $y = 13$ into the equation of the curve.

Work out the bracket

$1 - 3 = -2$.

Square the bracket

$(-2)^2 = 4$, so $a(-2)^2 = 4a$.

Subtract $5$ from both sides

$13 - 5 = 8$.

Divide by $4$

So the curve is $y = 2(x - 3)^2 + 5$.

Check the point ($1$, $13$)

The point ($1$, $13$) does lie on the curve.

Decide on the shape

The coefficient of the squared term is positive, so the curve is U-shaped with a minimum.

Use the fact that a square is never negative

The smallest the squared term can be is $0$.

State the turning point

The minimum point of the curve is ($3$, $5$).

Expand to check

Expand the bracket to write the curve in the usual form.

Multiply out

Multiply each term inside the bracket by $2$.

Simplify

$18 + 5 = 23$.

Check the line of symmetry

The line of symmetry is $x = 3$, which agrees with the turning point.

State the answer

The value of a is $2$ and the turning point is ($3$, $5$).