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Sagot :
Of course! Let's solve each equation step by step.
### (i) Solve the quadratic equation [tex]\(8x^2 + 2x - 15 = 0\)[/tex]
This is a standard quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
1. Identify coefficients:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -15\)[/tex]
2. Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (2)^2 - 4(8)(-15) = 4 + 480 = 484 \][/tex]
3. Find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting our values:
[tex]\[ x = \frac{-2 \pm \sqrt{484}}{2 \cdot 8} = \frac{-2 \pm 22}{16} \][/tex]
4. Calculate each root:
- For the positive case:
[tex]\[ x_1 = \frac{-2 + 22}{16} = \frac{20}{16} = 1.25 \][/tex]
- For the negative case:
[tex]\[ x_2 = \frac{-2 - 22}{16} = \frac{-24}{16} = -1.5 \][/tex]
Thus, the solutions for the equation [tex]\(8x^2 + 2x - 15 = 0\)[/tex] are:
[tex]\[ x = 1.25 \quad \text{and} \quad x = -1.5 \][/tex]
### (ii) Solve the quadratic equation [tex]\(x^2 - 16 = 0\)[/tex]
This can be rewritten as:
[tex]\[ x^2 = 16 \][/tex]
To solve for [tex]\(x\)[/tex], we take the square root of both sides of the equation.
1. Calculate the roots:
- The positive root:
[tex]\[ x_1 = \sqrt{16} = 4 \][/tex]
- The negative root:
[tex]\[ x_2 = -\sqrt{16} = -4 \][/tex]
Thus, the solutions for the equation [tex]\(x^2 - 16 = 0\)[/tex] are:
[tex]\[ x = 4 \quad \text{and} \quad x = -4 \][/tex]
In summary, the solutions are:
- For [tex]\(8x^2 + 2x - 15 = 0\)[/tex]: [tex]\(x = 1.25\)[/tex] and [tex]\(x = -1.5\)[/tex]
- For [tex]\(x^2 - 16 = 0\)[/tex]: [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex]
### (i) Solve the quadratic equation [tex]\(8x^2 + 2x - 15 = 0\)[/tex]
This is a standard quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
1. Identify coefficients:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -15\)[/tex]
2. Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (2)^2 - 4(8)(-15) = 4 + 480 = 484 \][/tex]
3. Find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting our values:
[tex]\[ x = \frac{-2 \pm \sqrt{484}}{2 \cdot 8} = \frac{-2 \pm 22}{16} \][/tex]
4. Calculate each root:
- For the positive case:
[tex]\[ x_1 = \frac{-2 + 22}{16} = \frac{20}{16} = 1.25 \][/tex]
- For the negative case:
[tex]\[ x_2 = \frac{-2 - 22}{16} = \frac{-24}{16} = -1.5 \][/tex]
Thus, the solutions for the equation [tex]\(8x^2 + 2x - 15 = 0\)[/tex] are:
[tex]\[ x = 1.25 \quad \text{and} \quad x = -1.5 \][/tex]
### (ii) Solve the quadratic equation [tex]\(x^2 - 16 = 0\)[/tex]
This can be rewritten as:
[tex]\[ x^2 = 16 \][/tex]
To solve for [tex]\(x\)[/tex], we take the square root of both sides of the equation.
1. Calculate the roots:
- The positive root:
[tex]\[ x_1 = \sqrt{16} = 4 \][/tex]
- The negative root:
[tex]\[ x_2 = -\sqrt{16} = -4 \][/tex]
Thus, the solutions for the equation [tex]\(x^2 - 16 = 0\)[/tex] are:
[tex]\[ x = 4 \quad \text{and} \quad x = -4 \][/tex]
In summary, the solutions are:
- For [tex]\(8x^2 + 2x - 15 = 0\)[/tex]: [tex]\(x = 1.25\)[/tex] and [tex]\(x = -1.5\)[/tex]
- For [tex]\(x^2 - 16 = 0\)[/tex]: [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex]
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