Question

determine the number of real solutions each quadratic equation has.
y = 12x² - 9x + 4
real solution(s)
4y - 7 = 5x² - x + 2 + 3y
real solution(s)
10x + y = -x² + 2
real solution(s)
y = (-x + 4)²
real solution(s)
done

Explanation:

To determine the number of real solutions for a quadratic equation (in the form \(ax^2 + bx + c = 0\) or a quadratic function \(y = ax^2 + bx + c\)), we use the discriminant \(D = b^2 - 4ac\). The rules are:

• If \(D > 0\), there are 2 real solutions.
• If \(D = 0\), there is 1 real solution (a repeated root).
• If \(D < 0\), there are 0 real solutions.

Equation 1: \(y = 12x^2 - 9x + 4\)

This is a quadratic function in the form \(y = ax^2 + bx + c\) where \(a = 12\), \(b = -9\), \(c = 4\).

Step 1: Calculate the discriminant

\(D = (-9)^2 - 4(12)(4)\)
\(D = 81 - 192\)
\(D = -111\)

Step 2: Determine the number of real solutions

Since \(D = -111 < 0\), there are 0 real solutions.

Equation 2: \(4y - 7 = 5x^2 - x + 2 + 3y\)

First, simplify the equation to solve for \(y\) and identify the quadratic in \(x\):

Step 1: Simplify the equation

\(4y - 3y = 5x^2 - x + 2 + 7\)
\(y = 5x^2 - x + 9\)
Now, the quadratic in \(x\) is \(5x^2 - x + 9\) (treating \(y\) as a function of \(x\)). Here, \(a = 5\), \(b = -1\), \(c = 9\).

Step 2: Calculate the discriminant

\(D = (-1)^2 - 4(5)(9)\)
\(D = 1 - 180\)
\(D = -179\)

Step 3: Determine the number of real solutions

Since \(D = -179 < 0\), there are 0 real solutions.

Equation 3: \(10x + y = -x^2 + 2\)

First, rearrange the equation to the form \(y = -x^2 - 10x + 2\). This is a quadratic function in \(x\) with \(a = -1\), \(b = -10\), \(c = 2\).

Step 1: Calculate the discriminant

\(D = (-10)^2 - 4(-1)(2)\)
\(D = 100 + 8\)
\(D = 108\)

Step 2: Determine the number of real solutions

Since \(D = 108 > 0\), there are 2 real solutions.

Equation 4: \(y = (-x + 4)^2\)

First, expand the right-hand side:

Step 1: Expand the square

\(y = (-x + 4)^2 = x^2 - 8x + 16\) (using \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = 4\), \(b = x\))
Now, the quadratic in \(x\) is \(x^2 - 8x + 16\) with \(a = 1\), \(b = -8\), \(c = 16\).

Step 2: Calculate the discriminant

\(D = (-8)^2 - 4(1)(16)\)
\(D = 64 - 64\)
\(D = 0\)

Step 3: Determine the number of real solutions

Since \(D = 0\), there is 1 real solution (a repeated root).

Answer:

1. \(y = 12x^2 - 9x + 4\): \(\boldsymbol{0}\) real solutions
2. \(4y - 7 = 5x^2 - x + 2 + 3y\): \(\boldsymbol{0}\) real solutions
3. \(10x + y = -x^2 + 2\): \(\boldsymbol{2}\) real solutions
4. \(y = (-x + 4)^2\): \(\boldsymbol{1}\) real solution