Answer :

We start with the equation
[tex]$$7x^2 + 15 = 26x.$$[/tex]

Step 1. Rewrite in standard form.
Subtract [tex]$26x$[/tex] from both sides to obtain
[tex]$$7x^2 - 26x + 15 = 0.$$[/tex]

Step 2. Identify the coefficients.
Here, we have
[tex]$$a = 7,\quad b = -26,\quad c = 15.$$[/tex]

Step 3. Compute the discriminant.
The discriminant is given by
[tex]$$\Delta = b^2 - 4ac.$$[/tex]
Substituting the coefficients, we get
[tex]$$\Delta = (-26)^2 - 4(7)(15) = 676 - 420 = 256.$$[/tex]

Step 4. Find the square root of the discriminant.
Since
[tex]$$\sqrt{256} = 16,$$[/tex]
this value will be used in the quadratic formula.

Step 5. Apply the quadratic formula.
The quadratic formula is
[tex]$$x = \frac{-b \pm \sqrt{\Delta}}{2a}.$$[/tex]
Substitute the known values:
[tex]$$x = \frac{-(-26) \pm 16}{2 \times 7} = \frac{26 \pm 16}{14}.$$[/tex]
This gives us two solutions:
- For the positive square root:
[tex]$$x_1 = \frac{26 + 16}{14} = \frac{42}{14} = 3.$$[/tex]
- For the negative square root:
[tex]$$x_2 = \frac{26 - 16}{14} = \frac{10}{14} = \frac{5}{7}.$$[/tex]

Final Answer:
The solutions to the equation are
[tex]$$x = 3 \quad \text{and} \quad x = \frac{5}{7}.$$[/tex]

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