Question

determining the nature of solutions
determine the number of real solutions each quadratic equation has
$y = 12x^2 - 3x + 4$
no real solution(s)
$4y - 1 = 5x^2 - x + 2 + 3y$
no real solution(s)
$10x + y = x^2 + 2$
real solution(s)
no
one
two
$y = (x + 4)^2$
one real solution(s)
done

Explanation:

Step1: Rearrange to standard quadratic form

For \(y=12x^2 - 9x + 4\), standard form: \(12x^2 - 9x + 4 = 0\)

Step2: Calculate discriminant \(D=b^2-4ac\)

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

Step3: Analyze discriminant (negative = no real roots)

Since \(D<0\), no real solutions.

Step4: Rearrange second equation to standard form

\(4y - 7 = 5x^2 - x + 2 + 3y\) simplifies to \(5x^2 - x + 9 - y = 0\), set \(y=0\) for roots: \(5x^2 - x + 9 = 0\)

Step5: Calculate discriminant

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

Step6: Analyze discriminant (negative = no real roots)

Since \(D<0\), no real solutions.

Step7: Rearrange third equation to standard form

\(10x + y = -x^2 + 2\) simplifies to \(x^2 + 10x + y - 2 = 0\), set \(y=0\) for roots: \(x^2 + 10x - 2 = 0\)

Step8: Calculate discriminant

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

Step9: Analyze discriminant (positive = two real roots)

Since \(D>0\), two real solutions.

Step10: Rearrange fourth equation to standard form

\(y=(x + 4)^2\) expands to \(x^2 + 8x + 16 - y = 0\), set \(y=0\) for roots: \(x^2 + 8x + 16 = 0\)

Step11: Calculate discriminant

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

Step12: Analyze discriminant (zero = one real root)

Since \(D=0\), one real solution.

Answer:

1. \(y=12x^2 - 9x + 4\): no real solution(s)
2. \(4y - 7 = 5x^2 - x + 2 + 3y\): no real solution(s)
3. \(10x + y = -x^2 + 2\): two real solution(s)
4. \(y=(x + 4)^2\): one real solution(s)