Rearranging Harder Formulae

Undo the addition and multiplication before taking the square root.

1. Undo the operations on x in reverse order. First subtract 5 from both sides: y-5=3x^2.
2. Divide both sides by 3: x^2=\frac{y-5}{3}.
3. Take the square root of both sides. Since x\ge0, use the positive root: x=\sqrt{\frac{y-5}{3}}.

Clear the denominator before isolating the term containing the subject.

1. Multiply by 5: 5p=2q-7.
2. Add 7 and divide by 2: q=\frac{5p+7}{2}.

Isolate the squared subject before taking the root.

1. Subtract 4as from both sides to get the squared term on its own: u^2=v^2-4as.
2. Take the square root of both sides. Since u\ge0, use the positive root: u=\sqrt{v^2-4as}.
3. Note that \sqrt{v^2-4as} is not v-\sqrt{4as}: you cannot square-root the terms separately.

Remove the term without the subject, then divide by its coefficient.

1. Subtract \pi r^2: \pi rs=A-\pi r^2.
2. Divide by \pi r: s=\frac{A-\pi r^2}{\pi r}.

Multiply by the denominator, then isolate the subject.

1. Multiply by (x+3): y(x+3)=4.
2. Expand and subtract 3y: xy=4-3y.
3. Divide by y: x=\frac{4-3y}{y}.

Factor out the subject because it occurs in both terms on the right.

1. Factor out p: q=p(7-3r).
2. Divide by (7-3r): p=\frac{q}{7-3r}.

Isolate the squared radius, remembering that a radius is positive.

1. Multiply both sides by 3 to clear the fraction: 3V=\pi r^2h.
2. Divide both sides by \pi h: r^2=\frac{3V}{\pi h}.
3. Take the square root. A radius is positive, so r=\sqrt{\frac{3V}{\pi h}}.

Square both sides before undoing the operations inside the root.

1. Square both sides: T^2=5x-2.
2. Add 2 and divide by 5: x=\frac{T^2+2}{5}.

Divide before squaring, so that the multiplier outside the root is handled correctly.

1. Divide both sides by a so the square root is on its own: \frac{p}{a}=\sqrt{q+b}.
2. Square both sides: \frac{p^2}{a^2}=q+b.
3. Subtract b: q=\frac{p^2}{a^2}-b.
4. This can also be written as a single fraction, by writing b as \frac{a^2b}{a^2}: q=\frac{p^2-a^2b}{a^2}.

Clear the fraction and then separate the two terms inside the brackets.

1. Multiply both sides by 2: 2A=h(a+b).
2. Expand the bracket: 2A=ah+bh.
3. Subtract ah from both sides: bh=2A-ah.
4. Divide both sides by h: b=\frac{2A-ah}{h}.

Clear the denominator and collect the terms containing the subject.

1. Multiply both sides by (a+b) to clear the fraction: K(a+b)=ab, so Ka+Kb=ab.
2. b appears in two terms. Get them on the same side: subtract Kb to give Ka=ab-Kb.
3. Take out the common factor b: Ka=b(a-K).
4. Divide both sides by (a-K): b=\frac{Ka}{a-K}.

After clearing the fraction, collect every term containing the subject on one side.

1. Multiply both sides by (x+2): y(x+2)=3x-4, so xy+2y=3x-4.
2. x appears in two terms. Collect them on one side: subtract xy and add 4 to both sides, giving 2y+4=3x-xy.
3. Take out the common factor x: 2y+4=x(3-y).
4. Divide both sides by (3-y): x=\frac{2y+4}{3-y}.

Square the entire multiplier as well as the square root.

1. Square both sides to remove the square root. Square the whole of the right-hand side, including 2\pi: T^2=4\pi^2\times\frac{L}{g}, that is T^2=\frac{4\pi^2L}{g}.
2. Multiply both sides by g: gT^2=4\pi^2L.
3. Divide both sides by T^2: g=\frac{4\pi^2L}{T^2}.

Square both sides before clearing the fraction.

1. Square both sides to remove the square root: p^2=\frac{x+5}{3}.
2. Multiply both sides by 3: 3p^2=x+5.
3. Subtract 5: x=3p^2-5.
4. x=\frac{p^2-5}{3} divides by 3 instead of multiplying, and (3p-5)^2 squares at the end instead of the start.

Clear the denominator, collect the terms containing the new subject and factorise.

Part (a)

1. Multiply both sides by (p-q): R(p-q)=p+q, so Rp-Rq=p+q.
2. p appears on both sides. Subtract p and add Rq to both sides: Rp-p=Rq+q.
3. Take out the common factors: p(R-1)=q(R+1).
4. Divide both sides by (R-1): p=\frac{q(R+1)}{R-1}.

Part (b)

1. Substitute into the formula from part (a): p=\frac{q(R+1)}{R-1}.
2. p=\frac{4\times(3+1)}{3-1}=\frac{16}{2}=8.

Expand after clearing the denominator and put every term containing the subject together.

1. Multiply both sides by (3x-4): y(3x-4)=2x+7, so 3xy-4y=2x+7.
2. x appears in two terms. Collect them on the left: subtract 2x and add 4y to both sides, giving 3xy-2x=4y+7.
3. Take out the common factor x: x(3y-2)=4y+7.
4. Divide both sides by (3y-2): x=\frac{4y+7}{3y-2}.

Remove the square root first, then collect the terms containing the subject.

1. Square both sides to remove the square root: y^2=\frac{4x-1}{x+2}.
2. Multiply both sides by (x+2): y^2(x+2)=4x-1, so xy^2+2y^2=4x-1.
3. Collect the x terms on one side: subtract xy^2 and add 1 to both sides, giving 2y^2+1=4x-xy^2.
4. Take out the common factor x: 2y^2+1=x(4-y^2).
5. Divide both sides by (4-y^2): x=\frac{2y^2+1}{4-y^2}.

Multiply both sides by ab to clear both fractions, then collect the terms containing x.

1. Multiply both sides by ab to clear both fractions: b(x+2)=a(x-3).
2. Expand the brackets: bx+2b=ax-3a.
3. Collect the x terms on one side and the other terms on the other: subtract bx and add 3a to both sides, giving 3a+2b=ax-bx.
4. Take out the common factor x: 3a+2b=x(a-b).
5. Divide both sides by (a-b): x=\frac{3a+2b}{a-b}.

Combine the two fractions that do not contain the subject before taking a reciprocal.

1. Subtract \frac{1}{u}: \frac{1}{v}=\frac{1}{f}-\frac{1}{u}.
2. Use denominator fu: \frac{1}{v}=\frac{u-f}{fu}.
3. Take reciprocals: v=\frac{fu}{u-f}.

Treat the square of the subject as one quantity until it has been isolated.

1. Multiply both sides by (x^2+4): y(x^2+4)=3x^2-2, so x^2y+4y=3x^2-2.
2. Treat x^2 as a single quantity. Collect the x^2 terms on one side: subtract x^2y and add 2 to both sides, giving 4y+2=3x^2-x^2y.
3. Take out the common factor x^2: 4y+2=x^2(3-y).
4. Divide both sides by (3-y): x^2=\frac{4y+2}{3-y}.
5. Since x\ge0, take the positive square root: x=\sqrt{\frac{4y+2}{3-y}}.