Compare the given quadratic equation 6x² - 23x + 21 = 0 with the standard form ax² + bx + c = 0. Here, a = 6, b = -23, and c = 21. We calculate the product a × c = 6 × 21 = 126.

Replace the middle term -23x with - 14x - 9x. Group the first two terms 6x² - 14x to factor out 2x, obtaining 2x(3x - 7). Group the remaining two terms -9x + 21 to factor out -3, obtaining -3(3x - 7).

Replace the middle term -23x with - 14x - 9x. Group the first two terms 6x² - 14x to factor out 2x, obtaining 2x(3x - 7). Group the remaining two terms -9x + 21 to factor out -3, obtaining -3(3x - 7).

By the zero product property, set each factor equal to zero: 2x - 3 = 0 gives x = (3)/(2), and 3x - 7 = 0 gives x = (7)/(3). Thus, the roots of the equation are (3)/(2) and (7)/(3).

Find the roots of the quadratic equation $6x^2 - 23x + 21 = 0$ by factorisation.

Answer: The roots of the quadratic equation are $x = \frac{3}{2}$ and $x = \frac{7}{3}$ .

Step-by-step solution

Step 1: Identify coefficients and calculate product

Compare the given quadratic equation $6x^2 - 23x + 21 = 0$ with the standard form $ax^2 + bx + c = 0$ . Here, $a = 6$ , $b = -23$ , and $c = 21$ . We calculate the product $a \times c = 6 \times 21 = 126$ .

Step 2: Find factors to split the middle term

We need two numbers $p$ and $q$ such that their product is $126$ and their sum is $-23$ . Factoring $126$ gives pairs such as $(-1, -126)$ , $(-2, -63)$ , $(-6, -21)$ , and $(-9, -14)$ . Since $-14 + (-9) = -23$ , the required numbers are $-14$ and $-9$ .

Step 3: Split the middle term and factor by grouping

Replace the middle term $-23x$ with - $14x - 9x$ . Group the first two terms $6x^2 - 14x$ to factor out $2x$ , obtaining $2x(3x - 7)$ . Group the remaining two terms $-9x + 21$ to factor out $-3$ , obtaining $-3(3x - 7)$ .

Step 4: Factor out the common binomial

Factor out the common binomial factor $(3x - 7)$ from both terms. This expresses the quadratic equation as a product of two linear factors: $(2x - 3)(3x - 7) = 0$ .

Step 5: Set each factor to zero and solve for x

By the zero product property, set each factor equal to zero: $2x - 3 = 0$ gives $x = \frac{3}{2}$ , and $3x - 7 = 0$ gives $x = \frac{7}{3}$ . Thus, the roots of the equation are $\frac{3}{2}$ and $\frac{7}{3}$ .

Solve your own maths question free →

Find the roots of the quadratic equation 6x2−23x+21=06x^2 - 23x + 21 = 06x2−23x+21=0 by factorisation.