If a number plus its reciprocal equals , find the number.

Understand the Question Let the unknown number be . Its reciprocal is (where ). The problem states that the sum of the number and its reciprocal is , giving the equation . By clearing the denominators, we convert this into a standard quadratic equation of the form and solve for by factorisation. Step 1 · Form the Quadratic Equation Let the required number be (). Given Multiplying both sides by Step 2 · Solve by Factorisation Splitting the middle term Setting each factor to zero Answer > or Common Mistakes Discarding One Root: Both and are valid solutions because the reciprocal of is and vice versa. Sign Errors in Middle-Term Splitting: Incorrectly splitting or making a sign error when factoring out from .

Exploring Algebraic Identities | EOT

Question 8

If a number plus its reciprocal equals $\dfrac{10}{3}$ , find the number.

Solution

Understand the Question

Let the unknown number be $x$ . Its reciprocal is $\dfrac{1}{x}$ (where $x \neq 0$ ).
The problem states that the sum of the number and its reciprocal is $\dfrac{10}{3}$ , giving the equation $x + \dfrac{1}{x} = \dfrac{10}{3}$ .
By clearing the denominators, we convert this into a standard quadratic equation of the form $ax^2 + bx + c = 0$ and solve for $x$ by factorisation.

Step 1 · Form the Quadratic Equation

Let the required number be $x$ ( $x \neq 0$ ).

Given $x + \dfrac{1}{x} = \dfrac{10}{3}$

Multiplying both sides by $3x$

\begin{aligned} 3x \left(x + \dfrac{1}{x}\right) &= 3x \left(\dfrac{10}{3}\right) \\[0.6em] 3x^2 + 3 &= 10x \\[0.6em] 3x^2 - 10x + 3 &= 0 \end{aligned}

Step 2 · Solve by Factorisation

Splitting the middle term $-10x = -9x - x$

\begin{aligned} 3x^2 - 9x - x + 3 &= 0 \\[0.6em] 3x(x - 3) - 1(x - 3) &= 0 \\[0.6em] (3x - 1)(x - 3) &= 0 \end{aligned}

Setting each factor to zero

\begin{aligned} 3x - 1 = 0 &\implies x = \dfrac{1}{3} \\[0.6em] x - 3 = 0 &\implies x = 3 \end{aligned}

Answer

$3$ or $\dfrac{1}{3}$

Common Mistakes

Discarding One Root: Both $3$ and $\dfrac{1}{3}$ are valid solutions because the reciprocal of $3$ is $\dfrac{1}{3}$ and vice versa.
Sign Errors in Middle-Term Splitting: Incorrectly splitting $-10x$ or making a sign error when factoring out $-1(x - 3)$ from $-x + 3$ .

More questions in EOT

Use suitable identities to find the following products:

(i) $(-3x + 4)^2$

(ii) $(2s + 7)(2s - 7)$

(iii) $\left(p^2 + \dfrac{1}{2}\right)\left(p^2 - \dfrac{1}{2}\right)$

(iv) $(2n + 7)(2n - 7)$

(v) $(s - 2t)(s^2 + 2st + 4t^2)$

(vi) $\left(\dfrac{1}{2r} - 4r\right)^2$

(vii) $(-3m + 4k - l)^2$

(viii) $\left(x - \dfrac{1}{3}y\right)^3$

(ix) $\left(\dfrac{7}{2}k - \dfrac{2}{3}m\right)^3$

Find the values using suitable identities:

(i) $17 \times 21$

(ii) $104 \times 96$

(iii) $24 \times 16$

(iv) $147^3$

(v) $199^3$

(vi) $127^3$

(vii) $(-107)^3$

(viii) $(-299)^3$

Factor the following algebraic expressions:

(i) $4y^2 + 1 + \dfrac{1}{16y^2}$

(ii) $9m^2 - \dfrac{1}{25n^2}$

(iii) $27b^3 - \dfrac{1}{64b^3}$

(iv) $x^2 + \dfrac{5x}{6} + \dfrac{1}{6}$

(v) $27u^3 - \dfrac{1}{125} - \dfrac{27u^2}{5} + \dfrac{9u}{25}$

(vi) $64y^3 + \dfrac{1}{125}z^3$

(vii) $p^3 + 27q^3 + r^3 - 9pqr$

(viii) $9m^2 - 12m + 4$

(ix) $9x^3 - \dfrac{8}{3}y^3 + \dfrac{z^3}{3} + 6xyz$

(x) $4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy$

(xi) $27u^3 - \dfrac{1}{216} - \dfrac{9u^2}{2} + \dfrac{u}{4}$

Simplify the following:

(i) $\dfrac{4x^2 + 4x + 1}{4x^2 - 1}$

(ii) $\dfrac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$

(iii) $\dfrac{s^3 + 125t^3}{s^2 - 2st - 35t^2}$

Note: Assume that the denominators are not equal to 0.

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.

(i) $6a^2 - 24b^2$

(ii) $3ps^2 - 15ps + 12p$

The village playground is shaped as a square of side 40 metres. A path of width $s$ metres is created around the playground for people to walk. Find an expression for the area of the path in terms of $s$ .

If a number plus its reciprocal equals $\dfrac{10}{3}$ , find the number.

A rectangular pool has area $2x^2 + 7x + 3$ square hastas. If its width is $2x + 1$ hastas, find its length. Hasta was a unit used to measure length.

Q10

If both $x - 2$ and $x - \dfrac{1}{2}$ are factors of $px^2 + 5x + r$ , show that $p = r$ .

Q11

If $a + b + c = 5$ and $ab + bc + ca = 10$ , then prove that $a^3 + b^3 + c^3 - 3abc = -25$ .

Q12

By factoring the expression, check that $n^3 - n$ is always divisible by 6 for all natural numbers $n$ . Give reasons.

Q13

Find the value of

(i) $x^3 + y^3 - 12xy + 64$ , when $x + y = -4$

(ii) $x^3 - 8y^3 - 36xy - 216$ , when $x = 2y + 6$