By the method of completing the square, obtain the roots of the following: x2 + 6x + 8 = 0 x2 - 4x − 5 = 0 x2…

Question

By the method of completing the square, obtain the roots of the following: x2 + 6x + 8 = 0 x2 - 4x − 5 = 0 x2 - 12x + 35 = 0 3x2 - 12x + 7 = 0 2x2 + 8x + 5 = 0

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Let's solve these quadratic equations using the method of completing the square. (f) (x² + 6x + 8 = 0) Step 1: Move the constant term to the right side of the equation. x² + 6x = −8; Step 2: Take half of the coefficient of (x) (which is (6)), square it (((6/2)² = 3² = 9)), and add it to both sides. x² + 6x + 9 = −8 + 9 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x + 3)² = 1 Step 4: Take the square root of both sides. x + 3 = ±√(1); x + 3 = ± 1 Step 5: Solve for (x). x = −3 ± 1; x₁ = −3 + 1 = −2; x₂ = −3 − 1 = −4 The roots are (−2, −4). --- (g) (x² − 4x − 5 = 0) Step 1: Move the constant term to the right side of the equation. x² − 4x = 5; Step 2: Take half of the coefficient of (x) (which is (−4)), square it (((−4/2)² = (−2)² = 4)), and add it to both sides. x² − 4x + 4 = 5 + 4 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x − 2)² = 9 Step 4: Take the square root of both sides. x − 2 = ±√(9); x − 2 = ± 3 Step 5: Solve for (x). x = 2 ± 3; x₁ = 2 + 3 = 5; x₂ = 2 − 3 = −1 The roots are (5, −1). --- (h) (x² − 12x + 35 = 0) Step 1: Move the constant term to the right side of the equation. x² − 12x = −35; Step 2: Take half of the coefficient of (x) (which is (−12)), square it (((−12/2)² = (−6)² = 36)), and add it to both sides. x² − 12x + 36 = −35 + 36 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x − 6)² = 1 Step 4: Take the square root of both sides. x − 6 = ±√(1); x − 6 = ± 1 Step 5: Solve for (x). x = 6 ± 1; x₁ = 6 + 1 = 7; x₂ = 6 − 1 = 5 The roots are (7, 5). --- (i) (3x² − 12x + 7 = 0) Step 1: Divide the entire equation by the coefficient of (x²) (which is (3)). x² − 4x + 7/3 = 0 Step 2: Move the constant term to the right side of the equation. x² − 4x = −7/3; Step 3: Take half of the coefficient of (x) (which is (−4)), square it (((−4/2)² = (−2)² = 4)), and add it to both sides. x² − 4x + 4 = −7/3 + 4; x² − 4x + 4 = −7/3 + 12/3; x² − 4x + 4 = 5/3 Step 4: Factor the left side as a perfect square trinomial. (x − 2)² = 5/3 Step 5: Take the square root of both sides. x − 2 = ±√(5/3); x − 2 = ±(√(5))/(√(3)); x − 2 = ±(√(5)√(3))/(3); x − 2 = ±(√(15))/(3) Step 6: Solve for (x). x = 2 ±(√(15))/(3); x = (6 ± √(15))/(3) The roots are ((6 ± √(15))/(3)). --- (j) (2x² + 8x + 5 = 0) Step 1: Divide the entire equation by the coefficient of (x²) (which is (2)). x² + 4x + 5/2 = 0 Step 2: Move the constant term to the right side of the equation. x² + 4x = −5/2; Step 3: Take half of the coefficient of (x) (which is (4)), square it (((4/2)² = 2² = 4)), and add it to both sides. x² + 4x + 4 = −5/2 + 4; x² + 4x + 4 = −5/2 + 8/2; x² + 4x + 4 = 3/2 Step 4: Factor the left side as a perfect square trinomial. (x + 2)² = 3/2 Step 5: Take the square root of both sides. x + 2 = ±√(3/2); x + 2 = ±(√(3))/(√(2)); x + 2 = ±(√(3)√(2))/(2); x + 2 = ±(√(6))/(2) Step 6: Solve for (x). x = −2 ±(√(6))/(2); x = (−4 ± √(6))/(2) The roots are ((−4 ± √(6))/(2)).

By the method of completing the square, obtain the roots of the following: x2 + 6x + 8 = 0 x2 − 4x − 5 = 0 x2 − 12x + 35 = 0 3x2 − 12x + 7 = 0 2x2 + 8x + 5 = 0

Answers

Step-by-step solution

Part f· 5 steps

Move the constant term to the right side of the equation.

Take half of the coefficient of $x$ (which is 6), square it ( $(6/2)^2 = 3^2 = 9$ ), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for $x$ .

Part g· 5 steps5, -1

Move the constant term to the right side of the equation.

Take half of the coefficient of $x$ (which is $-4$ ), square it ( $(-4/2)^2 = (-2)^2 = 4$ ), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for $x$ .

Part h· 5 steps7, 5

Move the constant term to the right side of the equation.

Take half of the coefficient of $x$ (which is $-12$ ), square it ( $(-12/2)^2 = (-6)^2 = 36$ ), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for $x$ .

Part h(i)· 12 steps
$\frac{6 \pm \sqrt{15}}{3}$

Divide the entire equation by the coefficient of $x^2$ (which is 3).

Move the constant term to the right side of the equation.

Take half of the coefficient of $x$ (which is $-4$ ), square it ( $(-4/2)^2 = (-2)^2 = 4$ ), and add it to both sides.

Factor the left side as a perfect square trinomial.

Take the square root of both sides.

Solve for $x$ .

Divide the entire equation by the coefficient of $x^2$ (which is 2).

Move the constant term to the right side of the equation.

Take half of the coefficient of $x$ (which is 4), square it ( $(4/2)^2 = 2^2 = 4$ ), and add it to both sides.

Factor the left side as a perfect square trinomial.

Take the square root of both sides.

Solve for $x$ .

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More Mathematics Questions

By the method of completing the square, obtain the roots of the following: x2 + 6x + 8 = 0 x2 − 4x − 5 = 0 x2 − 12x + 35 = 0 3x2 − 12x + 7 = 0 2x2 + 8x + 5 = 0

Answers

Step-by-step solution

Part f· 5 steps

Move the constant term to the right side of the equation.

Take half of the coefficient of xxx (which is 6), square it ((6/2)2=32=9(6/2)^2 = 3^2 = 9(6/2)2=32=9), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for xxx.

Part g· 5 steps5, -1

Move the constant term to the right side of the equation.

Take half of the coefficient of xxx (which is −4-4−4), square it ((−4/2)2=(−2)2=4(-4/2)^2 = (-2)^2 = 4(−4/2)2=(−2)2=4), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for xxx.

Part h· 5 steps7, 5

Move the constant term to the right side of the equation.

Take half of the coefficient of xxx (which is −12-12−12), square it ((−12/2)2=(−6)2=36(-12/2)^2 = (-6)^2 = 36(−12/2)2=(−6)2=36), and add it to both sides.

Factor the left side as a perfect square trinomial and simplify the right side.

Take the square root of both sides.

Solve for xxx.

Part h(i)· 12 steps6±153\frac{6 \pm \sqrt{15}}{3}36±15​​

Divide the entire equation by the coefficient of x2x^2x2 (which is 3).

Move the constant term to the right side of the equation.

Take half of the coefficient of xxx (which is −4-4−4), square it ((−4/2)2=(−2)2=4(-4/2)^2 = (-2)^2 = 4(−4/2)2=(−2)2=4), and add it to both sides.

Factor the left side as a perfect square trinomial.

Take the square root of both sides.

Solve for xxx.

Divide the entire equation by the coefficient of x2x^2x2 (which is 2).

Move the constant term to the right side of the equation.

Take half of the coefficient of xxx (which is 4), square it ((4/2)2=22=4(4/2)^2 = 2^2 = 4(4/2)2=22=4), and add it to both sides.

Factor the left side as a perfect square trinomial.

Take the square root of both sides.

Solve for xxx.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Take half of the coefficient of $x$ (which is 6), square it ( $(6/2)^2 = 3^2 = 9$ ), and add it to both sides.

Solve for $x$ .

Take half of the coefficient of $x$ (which is $-4$ ), square it ( $(-4/2)^2 = (-2)^2 = 4$ ), and add it to both sides.

Solve for $x$ .

Take half of the coefficient of $x$ (which is $-12$ ), square it ( $(-12/2)^2 = (-6)^2 = 36$ ), and add it to both sides.

Solve for $x$ .

Part h(i)· 12 steps
$\frac{6 \pm \sqrt{15}}{3}$

Divide the entire equation by the coefficient of $x^2$ (which is 3).

Take half of the coefficient of $x$ (which is $-4$ ), square it ( $(-4/2)^2 = (-2)^2 = 4$ ), and add it to both sides.

Solve for $x$ .

Divide the entire equation by the coefficient of $x^2$ (which is 2).

Take half of the coefficient of $x$ (which is 4), square it ( $(4/2)^2 = 2^2 = 4$ ), and add it to both sides.

Solve for $x$ .