Determine the nature of the roots for the following quadratic equations: a) 2x2 + 2x = 10 b) 2y - rac14yy+3…

Question

Determine the nature of the roots for the following quadratic equations: a) 2x2 + 2x = 10 b) 2y - rac14yy+3 + 1 = 0.

ScanSolve AI Tutor · Accepted Answer

To determine the nature of the roots of a quadratic equation ax² + bx + c = 0 without solving it, we use the discriminant, Δ = b² − 4ac. If Δ > 0, the roots are real and distinct. If Δ = 0, the roots are real and equal. If Δ < 0, the roots are complex (non-real) and distinct. a) The given equation is 2x² + 2x = 10. Step 1: Rewrite the equation in standard quadratic form ax² + bx + c = 0. 2x² + 2x − 10 = 0 Step 2: Identify the coefficients a, b, c. Here, a=2, b=2, c=−10. Step 3: Calculate the discriminant Δ = b² − 4ac. Δ = (2)² − 4(2)(−10); Δ = 4 − (−80); Δ = 4 + 80; Δ = 84 Step 4: Determine the nature of the roots based on the discriminant. Since Δ = 84 > 0, the roots are real and distinct. b) The given equation is 2y − (14y)/(y+3) + 1 = 0. Step 1: Clear the denominator by multiplying the entire equation by (y+3), assuming y ≠ −3. 2y(y+3) − 14y + 1(y+3) = 0; 2y² + 6y − 14y + y + 3 = 0; Step 2: Combine like terms to get the equation in standard quadratic form ay² + by + c = 0. 2y² − 7y + 3 = 0 Step 3: Identify the coefficients a, b, c. Here, a=2, b=−7, c=3. Step 4: Calculate the discriminant Δ = b² − 4ac. Δ = (−7)² − 4(2)(3); Δ = 49 − 24; Δ = 25 Step 5: Determine the nature of the roots based on the discriminant. Since Δ = 25 > 0, the roots are real and distinct. c) The given equation is (2x+3)² + 12 = 0. Step 1: Expand the squared term (2x+3)². (2x+3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9; Step 2: Substitute the expanded term back into the equation and simplify to standard quadratic form ax² + bx + c = 0. 4x² + 12x + 9 + 12 = 0; 4x² + 12x + 21 = 0 Step 3: Identify the coefficients a, b, c. Here, a=4, b=12, c=21. Step 4: Calculate the discriminant Δ = b² − 4ac. Δ = (12)² − 4(4)(21); Δ = 144 − 16(21); Δ = 144 − 336; Δ = −192 Step 5: Determine the nature of the roots based on the discriminant. Since Δ = −192 < 0, the roots are complex and distinct. d) The given equation is 6t² − 13t − 13 = 0. Step 1: The equation is already in standard quadratic form at² + bt + c = 0. Step 2: Identify the coefficients a, b, c. Here, a=6, b=−13, c=−13. Step 3: Calculate the discriminant Δ = b² − 4ac. Δ = (−13)² − 4(6)(−13); Δ = 169 − (−312); Δ = 169 + 312; Δ = 481 Step 4: Determine the nature of the roots based on the discriminant. Since Δ = 481 > 0, the roots are real and distinct.

Determine the nature of the roots for the following quadratic equations: a) 2x2 + 2x = 10 b) 2y − rac14yy+3 + 1 = 0.

Step-by-step solution

Part a· 4 steps

Rewrite the equation in standard quadratic form $ax^2 + bx + c = 0$ .

Identify the coefficients $a, b, c$.

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Determine the nature of the roots based on the discriminant.

Part b· 5 steps

Clear the denominator by multiplying the entire equation by $(y+3)$ , assuming $y \neq -3$ .

Combine like terms to get the equation in standard quadratic form $ay^2 + by + c = 0$ .

Identify the coefficients $a, b, c$.

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Determine the nature of the roots based on the discriminant.

Part c· 5 steps

Expand the squared term $(2x+3)^2$ .

Substitute the expanded term back into the equation and simplify to standard quadratic form $ax^2 + bx + c = 0$ .

Identify the coefficients $a, b, c$.

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Determine the nature of the roots based on the discriminant.

Part d· 4 steps

The equation is already in standard quadratic form $at^2 + bt + c = 0$ .

Identify the coefficients $a, b, c$.

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Determine the nature of the roots based on the discriminant.

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More Mathematics Questions

Determine the nature of the roots for the following quadratic equations: a) 2x2 + 2x = 10 b) 2y − rac14yy+3 + 1 = 0.

Step-by-step solution

Part a· 4 steps

Rewrite the equation in standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Identify the coefficients $a, b, c$.

Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.

Determine the nature of the roots based on the discriminant.

Part b· 5 steps

Clear the denominator by multiplying the entire equation by (y+3)(y+3)(y+3), assuming y≠−3y \neq -3y=−3.

Combine like terms to get the equation in standard quadratic form ay2+by+c=0ay^2 + by + c = 0ay2+by+c=0.

Identify the coefficients $a, b, c$.

Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.

Determine the nature of the roots based on the discriminant.

Part c· 5 steps

Expand the squared term (2x+3)2(2x+3)^2(2x+3)2.

Substitute the expanded term back into the equation and simplify to standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Identify the coefficients $a, b, c$.

Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.

Determine the nature of the roots based on the discriminant.

Part d· 4 steps

The equation is already in standard quadratic form at2+bt+c=0at^2 + bt + c = 0at2+bt+c=0.

Identify the coefficients $a, b, c$.

Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.

Determine the nature of the roots based on the discriminant.

Still stuck on this one?

More Mathematics Questions

Rewrite the equation in standard quadratic form $ax^2 + bx + c = 0$ .

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Clear the denominator by multiplying the entire equation by $(y+3)$ , assuming $y \neq -3$ .

Combine like terms to get the equation in standard quadratic form $ay^2 + by + c = 0$ .

Calculate the discriminant $\Delta = b^2 - 4ac$ .

Expand the squared term $(2x+3)^2$ .

Substitute the expanded term back into the equation and simplify to standard quadratic form $ax^2 + bx + c = 0$ .

Calculate the discriminant $\Delta = b^2 - 4ac$ .

The equation is already in standard quadratic form $at^2 + bt + c = 0$ .

Calculate the discriminant $\Delta = b^2 - 4ac$ .