Solve for x: 3.1 x²−2x−30, 3.2 x²+x−60, 3.6 10+x≤2x², 3.7 x²4

Question

Solve for x: 3.1 x²−2x−3>0, 3.2 x²+x−6<0, 3.3 x²−2x≥0, 3.4 x²+x−12≤0, 3.5 2-x-x²>0, 3.6 10+x≤2x², 3.7 x²>4

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3.1 x² − 2x − 3 > 0 Step 1: Find the roots of the quadratic equation x² − 2x − 3 = 0. (x−3)(x+1) = 0; x = 3 or x = −1; Step 2: Sketch the parabola y = x² − 2x − 3. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² − 2x − 3 > 0 means we are looking for the values of x where the parabola is above the x-axis. Solution: x < −1 or x > 3 3.2 x² + x − 6 < 0 Step 1: Find the roots of the quadratic equation x² + x − 6 = 0. (x+3)(x−2) = 0; x = −3 or x = 2; Step 2: Sketch the parabola y = x² + x − 6. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² + x − 6 < 0 means we are looking for the values of x where the parabola is below the x-axis. Solution: −3 < x < 2 3.3 x² − 2x ≥ 0 Step 1: Find the roots of the quadratic equation x² − 2x = 0. x(x−2) = 0; x = 0 or x = 2; Step 2: Sketch the parabola y = x² − 2x. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² − 2x ≥ 0 means we are looking for the values of x where the parabola is above or on the x-axis. Solution: x ≤ 0 or x ≥ 2 3.4 x² + x − 12 ≤ 0 Step 1: Find the roots of the quadratic equation x² + x − 12 = 0. (x+4)(x−3) = 0; x = −4 or x = 3; Step 2: Sketch the parabola y = x² + x − 12. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² + x − 12 ≤ 0 means we are looking for the values of x where the parabola is below or on the x-axis. Solution: −4 ≤ x ≤ 3 3.5 2 − x − x² > 0 Step 1: Rewrite the inequality by multiplying by −1 and reversing the sign. −x² − x + 2 > 0 x² + x − 2 < 0 Step 2: Find the roots of the quadratic equation x² + x − 2 = 0. (x+2)(x−1) = 0; x = −2 or x = 1; Step 3: Sketch the parabola y = x² + x − 2. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² + x − 2 < 0 means we are looking for the values of x where the parabola is below the x-axis. Solution: −2 < x < 1 3.6 10 + x ≤ 2x² Step 1: Rearrange the inequality to have 0 on one side. 0 ≤ 2x² − x − 10 2x² − x − 10 ≥ 0 Step 2: Find the roots of the quadratic equation 2x² − x − 10 = 0 using the quadratic formula x = (−b ± √(b² − 4ac))/(2a). x = (−(−1) ± √((−1)² − 4(2)(−10)))/(2(2)); x = (1 ± √(1 + 80))/(4); x = (1 ± √(81))/(4); x = (1 ± 9)/(4); x₁ = (1 + 9)/(4) = 10/4 = 5/2; x₂ = (1 − 9)/(4) = (−8)/(4) = −2; Step 3: Sketch the parabola y = 2x² − x − 10. Since the leading coefficient is positive, the parabola opens upwards. The inequality 2x² − x − 10 ≥ 0 means we are looking for the values of x where the parabola is above or on the x-axis. Solution: x ≤ −2 or x ≥ 5/2 3.7 x² > 4 Step 1: Rearrange the inequality to have 0 on one side. x² − 4 > 0 Step 2: Find the roots of the quadratic equation x² − 4 = 0. (x−2)(x+2) = 0; x = 2 or x = −2; Step 3: Sketch the parabola y = x² − 4. Since the leading coefficient is positive, the parabola opens upwards. The inequality x² − 4 > 0 means we are looking for the values of x where the parabola is above the x-axis. Solution: x < −2 or x > 2

Solve for x: 3.1 x²−2x−3>0, 3.2 x²+x−6<0, 3.3 x²−2x≥0, 3.4 x²+x−12≤0, 3.5 2−x−x²>0, 3.6 10+x≤2x², 3.7 x²>4

Answers

Step-by-step solution

Part 3.1· 2 steps

Find the roots of the quadratic equation $x^2 - 2x - 3 = 0$ .

Sketch the parabola $y = x^2 - 2x - 3$ .

Part 3.2· 2 steps
$-3 < x < 2$

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

Sketch the parabola $y = x^2 + x - 6$ .

Part 3.3· 2 steps
$x \le 0 \quad \text{or} \quad x \ge 2$

Find the roots of the quadratic equation $x^2 - 2x = 0$ .

Sketch the parabola $y = x^2 - 2x$ .

Part 3.4· 2 steps
$-4 \le x \le 3$

Find the roots of the quadratic equation $x^2 + x - 12 = 0$ .

Sketch the parabola $y = x^2 + x - 12$ .

Part 3.5· 3 steps
$-2 < x < 1$

Rewrite the inequality by multiplying by -1 and reversing the sign.

Find the roots of the quadratic equation $x^2 + x - 2 = 0$ .

Sketch the parabola $y = x^2 + x - 2$ .

Part 3.6· 3 steps
$x \le -2 \quad \text{or} \quad x \ge \frac{5}{2}$

Rearrange the inequality to have 0 on one side.

Find the roots of the quadratic equation $2x^2 - x - 10 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Sketch the parabola $y = 2x^2 - x - 10$ .

Part 3.7· 3 steps
$x < -2 \quad \text{or} \quad x > 2$

Rearrange the inequality to have 0 on one side.

Find the roots of the quadratic equation $x^2 - 4 = 0$ .

Sketch the parabola $y = x^2 - 4$ .

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More Mathematics Questions

Solve for x: 3.1 x²−2x−3>0, 3.2 x²+x−6<0, 3.3 x²−2x≥0, 3.4 x²+x−12≤0, 3.5 2−x−x²>0, 3.6 10+x≤2x², 3.7 x²>4

Answers

Step-by-step solution

Part 3.1· 2 steps

Find the roots of the quadratic equation x2−2x−3=0x^2 - 2x - 3 = 0x2−2x−3=0.

Sketch the parabola y=x2−2x−3y = x^2 - 2x - 3y=x2−2x−3.

Part 3.2· 2 steps−3<x<2-3 < x < 2−3<x<2

Find the roots of the quadratic equation x2+x−6=0x^2 + x - 6 = 0x2+x−6=0.

Sketch the parabola y=x2+x−6y = x^2 + x - 6y=x2+x−6.

Part 3.3· 2 stepsx≤0orx≥2x \le 0 \quad \text{or} \quad x \ge 2x≤0orx≥2

Find the roots of the quadratic equation x2−2x=0x^2 - 2x = 0x2−2x=0.

Sketch the parabola y=x2−2xy = x^2 - 2xy=x2−2x.

Part 3.4· 2 steps−4≤x≤3-4 \le x \le 3−4≤x≤3

Find the roots of the quadratic equation x2+x−12=0x^2 + x - 12 = 0x2+x−12=0.

Sketch the parabola y=x2+x−12y = x^2 + x - 12y=x2+x−12.

Part 3.5· 3 steps−2<x<1-2 < x < 1−2<x<1

Rewrite the inequality by multiplying by -1 and reversing the sign.

Find the roots of the quadratic equation x2+x−2=0x^2 + x - 2 = 0x2+x−2=0.

Sketch the parabola y=x2+x−2y = x^2 + x - 2y=x2+x−2.

Part 3.6· 3 stepsx≤−2orx≥52x \le -2 \quad \text{or} \quad x \ge \frac{5}{2}x≤−2orx≥25​

Rearrange the inequality to have 0 on one side.

Find the roots of the quadratic equation 2x2−x−10=02x^2 - x - 10 = 02x2−x−10=0 using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​.

Sketch the parabola y=2x2−x−10y = 2x^2 - x - 10y=2x2−x−10.

Part 3.7· 3 stepsx<−2orx>2x < -2 \quad \text{or} \quad x > 2x<−2orx>2

Rearrange the inequality to have 0 on one side.

Find the roots of the quadratic equation x2−4=0x^2 - 4 = 0x2−4=0.

Sketch the parabola y=x2−4y = x^2 - 4y=x2−4.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Find the roots of the quadratic equation $x^2 - 2x - 3 = 0$ .

Sketch the parabola $y = x^2 - 2x - 3$ .

Part 3.2· 2 steps
$-3 < x < 2$

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

Sketch the parabola $y = x^2 + x - 6$ .

Part 3.3· 2 steps
$x \le 0 \quad \text{or} \quad x \ge 2$

Find the roots of the quadratic equation $x^2 - 2x = 0$ .

Sketch the parabola $y = x^2 - 2x$ .

Part 3.4· 2 steps
$-4 \le x \le 3$

Find the roots of the quadratic equation $x^2 + x - 12 = 0$ .

Sketch the parabola $y = x^2 + x - 12$ .

Part 3.5· 3 steps
$-2 < x < 1$

Find the roots of the quadratic equation $x^2 + x - 2 = 0$ .

Sketch the parabola $y = x^2 + x - 2$ .

Part 3.6· 3 steps
$x \le -2 \quad \text{or} \quad x \ge \frac{5}{2}$

Find the roots of the quadratic equation $2x^2 - x - 10 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Sketch the parabola $y = 2x^2 - x - 10$ .

Part 3.7· 3 steps
$x < -2 \quad \text{or} \quad x > 2$

Find the roots of the quadratic equation $x^2 - 4 = 0$ .

Sketch the parabola $y = x^2 - 4$ .