Class 10 Mathematics – Algebra (Unit 2) Notes

A statement that relates two algebraic expressions using one of the symbols \(\le\) (less than or equal to), \(\ge\) (greater than or equal to), < (less than) or > (greater than). Unlike an equation, it almost always has a set of solutions, not just one value.

The replacement set is the set from which the variable can be chosen – typically \(x \in \mathbb{W}\) (whole numbers), \(x \in \mathbb{N}\), \(x \in \mathbb{I}\) (integers) or \(x \in \mathbb{R}\) (real numbers). The solution set is the set of all values from the replacement set that actually satisfy the inequation.

Example: for \(2x + 4 \le 14\) with \(x \in \mathbb{W}\), the solution set is \(\{0, 1, 2, 3, 4, 5\}\). With \(x \in \mathbb{R}\) the same inequation has infinitely many solutions: \(x \le 5\).

• The same number can be added to or subtracted from both sides.
• Both sides can be multiplied or divided by the same positive number – the sign stays the same.
• Both sides can be multiplied or divided by the same negative number – the sign must be reversed.

Also: you may not multiply or divide by zero, and you may not multiply or divide by an unknown whose sign you do not know.

Use a closed (filled) dot for \(\le\) or \(\ge\) (the endpoint is included) and an open (hollow) dot for < or > (the endpoint is excluded). Shade the line in the direction of the solution set.

An equation of the form \(ax^2 + bx + c = 0\) where \(a \ne 0\). The values of \(x\) that satisfy it are called its roots (or solutions). A quadratic equation has at most two real roots.

Express \(ax^2 + bx + c\) as a product of two linear factors. If the product is zero, then one of the factors must be zero (the zero-product rule). For example \(x^2 - 5x + 6 = (x - 2)(x - 3) = 0\) gives \(x = 2\) or \(x = 3\).

For the equation \(px^2 - qx + r = 0\) (here \(a = p\), \(b = -q\), \(c = r\)), the roots are real and equal if and only if the discriminant is zero: \[b^2 - 4ac = (-q)^2 - 4pr = q^2 - 4pr = 0 \;\Rightarrow\; q^2 = 4pr.\]

For \(ax^2 + bx + c = 0\), the discriminant is \(D = b^2 - 4ac\):

• \(D > 0\) – two distinct real roots;
• \(D = 0\) – real and equal roots;
• \(D < 0\) – no real roots.

If the roots are required "correct to one decimal place" or "to 3 significant figures", use the quadratic formula and round only at the end.

The ratio \(a : b\) is the comparison of two quantities of the same kind, written as a fraction \(\frac{a}{b}\) in its lowest terms. Example: the ratio 12 : 8 equals 3 : 2.

Four numbers \(a, b, c, d\) are in proportion when \(a : b = c : d\), written \(a : b :: c : d\). The key property is the cross product:

\[a : b = c : d \quad \Leftrightarrow \quad ad = bc.\]

• Mean proportional between \(a\) and \(b\): a number \(x\) such that \(a : x = x : b\), i.e. \(x^2 = ab\), so \(x = \sqrt{ab}\).
• Third proportional to \(a\) and \(b\): a number \(x\) such that \(a : b = b : x\), so \(x = \frac{b^2}{a}\).
• Fourth proportional to \(a, b, c\): a number \(x\) such that \(a : b = c : x\), so \(x = \frac{bc}{a}\).

If \(a, b, c\) are in continued proportion, then \(b^2 = ac\).

If \(\dfrac{a}{b} = \dfrac{c}{d}\), then \(\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}\). This rule lets you "compose" and "divide" numerators and denominators together and is very useful for solving equations with surds.

If a polynomial \(P(x)\) is divided by \(x - a\), the remainder is \(P(a)\). You never need to divide: just substitute \(x = a\) into the polynomial.

\(x - a\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). This follows from the remainder theorem when the remainder is zero.

If \(x - a\) is a factor of a polynomial \(P(x)\) that contains an unknown coefficient \(k\), substitute \(x = a\), set \(P(a) = 0\), and solve the resulting linear equation for \(k\).

Test small integer values (\(x = \pm 1, \pm 2, \pm 3, \ldots\)) to find one linear factor. Divide to obtain a quadratic, then factorise the quadratic (or use the quadratic formula) to get all three factors.

A matrix is a rectangular arrangement of numbers in rows and columns. Its order is \(m \times n\) (read "\(m\) by \(n\)"), where \(m\) = number of rows and \(n\) = number of columns. A matrix with one column is a column matrix; with one row, a row matrix.

Two matrices are equal when they have the same order and every corresponding element is equal. This allows you to compare entries and solve for unknowns. Note that \(\begin{bmatrix}1 & 2\end{bmatrix} \ne \begin{bmatrix}1 \\ 2\end{bmatrix}\) because the orders differ.

Matrices of the same order are added (or subtracted) element by element. In scalar multiplication, every element is multiplied by the scalar: \(k\begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{bmatrix}ka & kb \\ kc & kd\end{bmatrix}\).

\(AB\) exists only when the number of columns of \(A\) equals the number of rows of \(B\). If \(A\) is \(m \times q\) and \(B\) is \(q \times n\), then \(AB\) is \(m \times n\), with each entry the "row × column" dot product. Matrix multiplication is not commutative in general.

The identity matrix is a square matrix with 1 on the main diagonal and 0 elsewhere, e.g. \(I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\). For any compatible matrix \(A\), \(AI = IA = A\).

A sequence in which each term after the first differs from the previous term by a fixed number called the common difference \(d\). Example: \(15, 30, 45, 60, \ldots\) is an AP with \(a = 15\) and \(d = 15\).

\(T_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. To find a middle term given two other terms, set up equations in \(a\) and \(d\).

\(S_n = \dfrac{n}{2}\big[2a + (n-1)d\big]\), or \(S_n = \dfrac{n}{2}(a + l)\) where \(l = T_n\) is the last term. Both give the same total; use the second form when the last term is known.

If \(a, b, c\) are in AP, then \(2b = a + c\) (the middle term is the average of its neighbours). This is a quick checking tool.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed number called the common ratio \(r\). Example: \(15, 30, 60, 120, \ldots\) is a GP with \(a = 15\), \(r = 2\).

\(T_n = ar^{n-1}\). The common ratio \(r = \frac{T_{k+1}}{T_k}\). A ratio \(r\) may be negative or fractional (e.g. \(1, -\frac12, \frac14, \ldots\)).

\(S_n = \dfrac{a(r^n - 1)}{r - 1}\). For \(|r| < 1\) the same formula is often written \(S_n = \dfrac{a(1 - r^n)}{1 - r}\), which keeps the denominator positive. Both are identical.

\(a, b, c\) are in GP if and only if \(b^2 = ac\) (the middle term squared equals the product of its neighbours).