Question

use the quadratic formula to solve for x.
$6x^2 - 7x - 2 = 0$
(if there is more than one solution, separate them with commas.)
$x = \square$

Explanation:

Step1: Identify a, b, c

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 6\), \(b=-7\), \(c = -2\).

Step2: Quadratic formula

Quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute values:
First, calculate discriminant \(D=b^2 - 4ac=(-7)^2-4\times6\times(-2)=49 + 48 = 97\)? Wait, no, wait: \((-7)^2=49\), \(4\times6\times2 = 48\), but \(c=-2\), so \(-4ac=-4\times6\times(-2)=48\). So \(D = 49 + 48 = 97\)? Wait, no, wait the equation is \(6x^2-7x - 2 = 0\), so \(a = 6\), \(b=-7\), \(c=-2\). Then \(b^2-4ac=(-7)^2-4\times6\times(-2)=49 + 48 = 97\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \(6x^2-7x - 2 = 0\), let's check discriminant again: \(b^2 - 4ac = (-7)^2 - 4*6*(-2)=49 + 48 = 97\)? Wait, but 97 is a prime? Wait, no, maybe I miscalculated. Wait, 6x² -7x -2. Let's factor it: 6x² -8x +x -2 = 2x(3x -4)+1(3x -4)=(2x +1)(3x -4)=0. So roots are x=-1/2 and x=4/3. Wait, so my discriminant calculation was wrong? Wait, no: \(b^2 -4ac = (-7)^2 -4*6*(-2)=49 + 48 = 97\)? But (2x +1)(3x -4)=6x² -8x +3x -4=6x² -5x -4. No, that's not the original equation. Wait, original equation is 6x² -7x -2. Let's do discriminant again: \(a=6\), \(b=-7\), \(c=-2\). So \(b^2 -4ac = (-7)^2 -4*6*(-2)=49 + 48 = 97\)? Wait, but 97 is not a perfect square. But when I factor, let's try again: 6x² -7x -2. Multiply a and c: 6*(-2)=-12. Find two numbers that multiply to -12 and add to -7. The numbers are -8 and +1. So 6x² -8x +x -2 = 2x(3x -4)+1(3x -4)? No, 6x² -8x +x -2 = 2x(3x -4)+1(x - 2). No, that's wrong. Wait, 6x² -7x -2. Let's use quadratic formula correctly. \(x=\frac{7\pm\sqrt{49 + 48}}{12}=\frac{7\pm\sqrt{97}}{12}\)? But that contradicts factoring. Wait, no, I must have factored wrong. Wait, (3x - 4)(2x + 1)=6x² +3x -8x -4=6x² -5x -4. Not the original. Wait, original is 6x² -7x -2. So let's do quadratic formula again: \(a=6\), \(b=-7\), \(c=-2\). So \(x=\frac{7\pm\sqrt{(-7)^2 -4*6*(-2)}}{2*6}=\frac{7\pm\sqrt{49 + 48}}{12}=\frac{7\pm\sqrt{97}}{12}\)? But that's not matching factoring. Wait, I think I messed up the sign of c. Wait, the equation is 6x² -7x -2 = 0, so c=-2. So -4ac = -4*6*(-2)=48. So discriminant is 49 + 48 = 97. But 97 is a prime, so roots are (7 + sqrt(97))/12 and (7 - sqrt(97))/12? But that can't be, because when I plug x=4/3 into 6x² -7x -2: 6*(16/9) -7*(4/3)-2= 32/3 -28/3 -2=4/3 -2= -2/3 ≠0. Wait, so my factoring was wrong. Wait, let's do it again. 6x² -7x -2. Let's use quadratic formula: \(x=\frac{7\pm\sqrt{49 + 48}}{12}=\frac{7\pm\sqrt{97}}{12}\). Wait, but 97 is approximately 9.849. So (7 + 9.849)/12≈16.849/12≈1.404, (7 -9.849)/12≈-2.849/12≈-0.237. But when I plug x=-0.237 into 6x² -7x -2: 6*(0.056)-7*(-0.237)-2≈0.336 +1.659 -2≈0. So that works. And x≈1.404: 6*(1.971)-7*(1.404)-2≈11.826 -9.828 -2≈-0.002, which is due to approximation. So my initial factoring attempt was wrong. So let's proceed with quadratic formula.

Wait, but the problem is to use quadratic formula. So:

\(x=\frac{-(-7)\pm\sqrt{(-7)^2 -4*6*(-2)}}{2*6}=\frac{7\pm\sqrt{49 + 48}}{12}=\frac{7\pm\sqrt{97}}{12}\)? Wait, but 49 + 48 is 97? Wait, 4*6*2 is 48, but since c is -2, -4ac is -4*6*(-2)=48. So yes, discriminant is 49 + 48 = 97. So the roots are \(\frac{7 + \sqrt{97}}{12}\) and \(\frac{7 - \sqrt{97}}{12}\). Wait, but that seems complicated. Wait, maybe I made a mistake in the equation. Wait, the equation is 6x² -7x -2 = 0. Let me check with quadratic formula again.

Wait, no, wait: 6x² -7x -2. Let's compute discriminant again: b² -4ac = (-7)² -4*6*(-2) = 49 + 48 = 97. Yes. So the roots are [7 ± √97]/12. But that's approximately (7 + 9.849)/12 ≈16.849/12≈1.404, and (7 -9.849)/12≈-2.849/12≈-0.237. But maybe the problem has a typo, or I misread the equation. Wait, maybe the equation is 6x² -7x -3 = 0? No, the user wrote 6x² -7x -2 = 0.

Wait, but let's check again. Wait, 6x² -7x -2. Let's use quadratic formula:

\(x = \frac{7 \pm \sqrt{49 + 48}}{12} = \frac{7 \pm \sqrt{97}}{12}\). But √97 is irrational. So the solutions are \(\frac{7 + \sqrt{97}}{12}\) and \(\frac{7 - \sqrt{97}}{12}\). But maybe I made a mistake in the sign of b. Wait, the equation is 6x² -7x -2 = 0, so b is -7, so -b is 7. Correct.

Wait, but the initial factoring attempt was wrong. So the correct roots using quadratic formula are \(\frac{7 + \sqrt{97}}{12}\) and \(\frac{7 - \sqrt{97}}{12}\). But let's check with calculator: √97 ≈9.849. So 7 + 9.849 = 16.849, divided by 12 ≈1.404. 7 -9.849 = -2.849, divided by 12 ≈-0.237. So those are the roots.

But maybe the problem was 6x² -8x -2 = 0? No, the user wrote -7x. So proceed with the given equation.

So step by step:

1. Identify a, b, c: a=6, b=-7, c=-2.

2. Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 -4ac}}{2a}\).

3. Substitute: \(x=\frac{7\pm\sqrt{(-7)^2 -4*6*(-2)}}{12}=\frac{7\pm\sqrt{49 + 48}}{12}=\frac{7\pm\sqrt{97}}{12}\).

Answer:

\(\frac{7 + \sqrt{97}}{12}, \frac{7 - \sqrt{97}}{12}\)

Wait, but wait, I just realized that maybe I made a mistake in discriminant calculation. Wait, 6x² -7x -2. Let's check with another method. Let's use completing the square.

6x² -7x = 2

x² - (7/6)x = 2/6 = 1/3

x² - (7/6)x + (7/12)² = 1/3 + (49/144)

(x - 7/12)² = (48/144 + 49/144) = 97/144

x - 7/12 = ±√(97)/12

x = (7 ± √97)/12. Yes, so that's correct. So the roots are (7 + √97)/12 and (7 - √97)/12.