How many real-number solutions does the equation have?
–7x2 + 6x + 3 = 0
one solution
two solutions***********
no solutions
infinitely many solutions
you are correct; the discriminant is positive.
Well, I don't have a calculator so I can't give you the exact answer, but I can tell you one thing - that equation seems like it's having a mood swing. It's got both positive and negative numbers, and it's asking for solutions. It's like it can't make up its mind! So, it's possible that it has two solutions, one for when it's feeling positive and one for when it's feeling negative. Just like a rollercoaster of answers!
To determine the number of real-number solutions for the equation -7x^2 + 6x + 3 = 0, we can use the discriminant. The discriminant, denoted by Δ, is calculated using the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, a = -7, b = 6, and c = 3. Plugging these values into the formula, we get Δ = 6^2 - 4(-7)(3).
Calculating further, Δ = 36 - (-84) = 36 + 84 = 120.
Now, the number of real-number solutions depends on the discriminant Δ:
- If Δ > 0, then the equation has two distinct real solutions.
- If Δ = 0, then the equation has one real solution.
- If Δ < 0, then the equation has no real solutions.
- If the equation is linear (a = 0), then it has one real solution.
In this case, since Δ = 120 > 0, the equation -7x^2 + 6x + 3 = 0 has two real solutions. Therefore, the correct answer is "two solutions."
To find the number of real-number solutions of the equation -7x^2 + 6x + 3 = 0, we can use the discriminant (denoted by D) from the quadratic formula.
The quadratic formula states that given an equation of the form Ax^2 + Bx + C = 0, the solutions are given by:
x = (-B ± √(B^2 - 4AC)) / 2A
In this case, A = -7, B = 6, and C = 3. The discriminant, which is the expression inside the square root, is given by B^2 - 4AC:
D = 6^2 - 4*(-7)*(3)
= 36 + 84
= 120
Now, we can determine the number of real-number solutions based on the value of the discriminant.
- If the discriminant is positive (D > 0), then there are two distinct real solutions.
- If the discriminant is zero (D = 0), then there is one real solution.
- If the discriminant is negative (D < 0), then there are no real solutions.
In this case, the discriminant D = 120, which is positive. Therefore, the equation -7x^2 + 6x + 3 = 0 has two real solutions.