Leaving Cert Mathematics Logs and Indices: Given that $f(x) = 3x^2 + 8x

Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question (b) - 2021

Question (b)

$Given-that-$f(x)-=-3x^2-+-8x---35$,-where-$x-\in-\mathbb{R}$,-find-the-two-roots-of-$f(x)-=-0$-Leaving Cert Mathematics-Question (b)-2021.png$

Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$. Hence or otherwise, solve the equation $3^{2m+1} = 35 - 8(3^m)$, whe... show full transcript

Worked Solution & Example Answer:Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question (b) - 2021

Step 1

Find the two roots of $f(x) = 0$

96%

114 rated

Answer

To find the roots of the equation $f(x) = 3x^2 + 8x - 35 = 0$ , we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 3$ , $b = 8$ , and $c = -35$ . Plugging these values into the formula gives:

$x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 3 \cdot (-35)}}{2 \cdot 3}$

Calculating the discriminant:

$b^2 - 4ac = 64 + 420 = 484$

Thus,

$x = \frac{-8 \pm \sqrt{484}}{6} = \frac{-8 \pm 22}{6}$

This provides us with two potential solutions:

$x = \frac{14}{6} = \frac{7}{3}$
$x = \frac{-30}{6} = -5$

So, the two roots are $x = \frac{7}{3}$ and $x = -5$ .

Step 2

Solve the equation $3^{2m+1} = 35 - 8(3^m)$

99%

104 rated

Answer

We start with the equation:

$3^{2m+1} = 35 - 8(3^m)$

Using the property $3^{2m} = (3^m)^2$ , we can rewrite:

$3^{2m}(3) = 35 - 8(3^m)$

Letting $x = 3^m$ , we rewrite the equation as:

$3x^2 + 8x - 35 = 0$

Factoring this results in:

$(3x - 5)(x + 7) = 0$

This gives the solutions:

$x = \frac{5}{3}$ (thus, $3^m = \frac{5}{3}$ )
$x = -7$ (not applicable since $3^m > 0$ )

Taking the valid solution: $3^m = \frac{5}{3}$ Taking logarithms:

$m = \log_3(\frac{5}{3})$

To express this in the required form $m = \log_b p - q$ , we can rewrite:

$m = \log_3 5 - \log_3 3 = \log_3 5 - 1$

Thus, we have $p = 5$ , $q = 1$ , and $b = 3$ .

Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question (b) - 2021

Worked Solution & Example Answer:Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question (b) - 2021

Find the two roots of $f(x) = 0$

Solve the equation $3^{2m+1} = 35 - 8(3^m)$

Join the Leaving Cert students using SimpleStudy...