Finding Values for a Perfect Square Expression

We are asked to find the values of \(m\) for which the expression \(mx^2 + mx + 8x + 9\) is a perfect square. A perfect square quadratic expression is one that can be written in the form \((px+q)^2\) or \((px-q)^2\).

Structuring the Quadratic Expression

First, let's rewrite the given expression by grouping the terms with \(x\):

The expression is \(mx^2 + mx + 8x + 9\).

Grouping the \(x\) terms, we get:

\(mx^2 + (m+8)x + 9\)

This is a quadratic expression in the standard form \(Ax^2 + Bx + C\), where:

• \(A = m\)
• \(B = m+8\)
• \(C = 9\)

Condition for a Perfect Square Quadratic

For a quadratic expression \(Ax^2 + Bx + C\) to be a perfect square of a linear term (like \((px+q)^2\) or \((px-q)^2\)), the following conditions must be met:

• The coefficient of \(x^2\), \(A\), must be a perfect square (or \(A=0\), but then it's not a quadratic).
• The constant term, \(C\), must be a perfect square.
• The discriminant of the quadratic equation \(Ax^2 + Bx + C = 0\) must be zero. The discriminant is given by the formula \(\Delta = B^2 - 4AC\).

In our expression, \(A=m\) and \(C=9\). \(C=9\) is a perfect square (\(3^2\)). For the expression to be a perfect square of the form \((px+q)^2\), \(A\) must also be a perfect square. However, the condition for the discriminant being zero is sufficient for the expression to be a perfect square *polynomial* (which is what is generally implied in such questions for quadratic form). If \(A=0\), the expression becomes a linear term plus a constant (\(8x+9\)), which is not a perfect square unless \(8=0\) and \(9\) is a perfect square, which is not the case here.

Therefore, we focus on the discriminant condition: \(\Delta = B^2 - 4AC = 0\).

Calculating the Discriminant

Substitute the values of \(A\), \(B\), and \(C\) into the discriminant formula:

\(\Delta = (m+8)^2 - 4(m)(9)\)

\(\Delta = (m^2 + 16m + 64) - 36m\)

\(\Delta = m^2 + 16m - 36m + 64\)

\(\Delta = m^2 - 20m + 64\)

Solving for m

For the expression to be a perfect square, the discriminant must be zero. So, we set the calculated discriminant equal to zero:

\(m^2 - 20m + 64 = 0\)

This is a quadratic equation in terms of \(m\). We can solve this by factoring, using the quadratic formula, or completing the square. Factoring is a common method if the factors are easy to find.

We need two numbers that multiply to \(+64\) and add up to \(-20\). These numbers are \(-4\) and \(-16\), because \((-4) \times (-16) = 64\) and \((-4) + (-16) = -20\).

So, we can factor the quadratic equation as:

\((m - 4)(m - 16) = 0\)

For this product to be zero, one or both of the factors must be zero.

• Case 1: \(m - 4 = 0 \implies m = 4\)
• Case 2: \(m - 16 = 0 \implies m = 16\)

These are the two possible values for \(m\).

Checking the Solutions

Let's verify if for these values of \(m\), the original expression is indeed a perfect square.

If \(m = 4\):

The expression becomes \(4x^2 + (4+8)x + 9 = 4x^2 + 12x + 9\).

We can recognize this as \((2x)^2 + 2(2x)(3) + (3)^2\), which fits the form \((a+b)^2 = a^2 + 2ab + b^2\). So, \(4x^2 + 12x + 9 = (2x+3)^2\). This is a perfect square.

If \(m = 16\):

The expression becomes \(16x^2 + (16+8)x + 9 = 16x^2 + 24x + 9\).

We can recognize this as \((4x)^2 + 2(4x)(3) + (3)^2\), which fits the form \((a+b)^2 = a^2 + 2ab + b^2\). So, \(16x^2 + 24x + 9 = (4x+3)^2\). This is also a perfect square.

Both values, \(m=4\) and \(m=16\), make the expression a perfect square.

Conclusion

The values of \(m\) for which the expression \(mx^2 + mx + 8x + 9\) is a perfect square are 4 and 16.

Revision Table: Perfect Square Quadratics

Additional Information: Discriminant and Roots

The discriminant (\(\Delta = B^2 - 4AC\)) of a quadratic equation \(Ax^2 + Bx + C = 0\) tells us about the roots:

• If \(\Delta > 0\), there are two distinct real roots. The quadratic can be factored into \((x-r_1)(x-r_2)\) where \(r_1 \ne r_2\).
• If \(\Delta = 0\), there is exactly one real root (a repeated root). The quadratic can be factored into \((x-r)^2\), which is a perfect square.
• If \(\Delta < 0\), there are no real roots (two complex conjugate roots). The quadratic cannot be factored into linear terms with real coefficients.

When a quadratic expression \(Ax^2 + Bx + C\) is a perfect square, it means the corresponding quadratic equation \(Ax^2 + Bx + C = 0\) has exactly one root (or two equal roots). This happens precisely when the discriminant is zero.

In our problem, by setting the discriminant equal to zero (\(m^2 - 20m + 64 = 0\)), we found the values of \(m\) that cause the expression \(mx^2 + (m+8)x + 9\) to have a single root when set to zero, thus making the expression a perfect square polynomial.