Algebra: Concepts, Formulas, and Clear Examples

Algebra MCQs reward identities, factorization, and clean substitution—not brute expansion. Scan for $(a\pm b)^2$, $a^2-b^2$, and hidden quadratics before doing long arithmetic.

Expect linear word problems, quadratic roots/discriminant, simultaneous equations, and occasional modulus or integer constraints on $x$.

When options are numeric, backsolve from the middle choice after estimating magnitude.

Linear equations

Solve linear

$$ax + b = c \;\Rightarrow\; x = \frac{c-b}{a}$$

• Consecutive integers: $n, n+1$; consecutive even: $2n, 2n+2$.
• Translate each sentence into one equation before combining.

Example — Three consecutive sum

Sum of three consecutive integers is 72. Let $n-1, n, n+1$. Sum = $3n=72$ → $n=24$. Largest = 25.

Identities for speed

Square of sum

$$(a+b)^2 = a^2 + 2ab + b^2$$

Difference of squares

$$a^2 - b^2 = (a-b)(a+b)$$

Product from sum/diff

$$4ab = (a+b)^2 - (a-b)^2$$

Quadratics

Roots

$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

Sum/product

$$\alpha+\beta=-\frac{b}{a},\ \alpha\beta=\frac{c}{a}$$

Equal roots

$$b^2 - 4ac = 0$$

Example — Find k for equal roots

$x^2 + kx + 16 = 0$ has equal roots → $k^2 - 64 = 0$ → $k = \pm 8$.

Factorization and cancellation

• Factor before canceling; never cancel across $+$.
• Note values that make denominator zero.

Inequalities

AM ≥ GM

$$\frac{x+y}{2} \ge \sqrt{xy}\quad (x,y>0)$$

• Multiplying/dividing by negative flips $<$ to $>$.
• Count integer solutions carefully on inclusive bounds.

Simultaneous linear systems

Two lines

$$a_1x+b_1y=c_1,\ a_2x+b_2y=c_2$$

Eliminate or substitute to one variable.

Step-by-step method for exam questions

1. Look for identities/factoring patterns.
2. Define variables; write equations from wording.
3. Simplify rational expressions; track exclusions.
4. Solve (linear / quadratic / system).
5. Filter roots by context (positive length, integer, etc.).
6. Verify with option or substitution.

Formula cheat sheet

Frequently asked questions

Backsolve order?

Try (B) or (C) first when options ascend numerically.

Hidden quadratic?

Try $t=x^2$ or symmetric $x+1/x$ forms.

Can roots be complex?

If options are real-only, $\Delta<0$ may be the intended conclusion.

Cancel $(x-2)$?

Only if $x\neq 2$; MCQ may test excluded value separately.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.