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Use the quadratic formula to find the exact solution to the given equation. (Enter your answers as a comma-separated list.)

[tex]\[ 7x^2 - 9x + 1 = 0 \][/tex]

Provide solutions accurate to two decimal places.

Smaller [tex]\( x \)[/tex]-value \quad [tex]\( x = \)[/tex] [tex]\(\square\)[/tex]

Larger [tex]\( x \)[/tex]-value \quad [tex]\( x = \)[/tex] [tex]\(\square\)[/tex]


Sagot :

To solve the quadratic equation [tex]\(7x^2 - 9x + 1 = 0\)[/tex] using the quadratic formula, we follow a systematic approach. The quadratic formula is:

[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]

Here, the coefficients are:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -9\)[/tex]
- [tex]\(c = 1\)[/tex]

Step-by-step, let's substitute these values into the quadratic formula.

1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

Substituting [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:

[tex]\[ \Delta = (-9)^2 - 4 \cdot 7 \cdot 1 = 81 - 28 = 53 \][/tex]

2. Calculate the Two Solutions:
Using the quadratic formula, we find two solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:

[tex]\[ x = \frac{{-b \pm \sqrt{\Delta}}}{2a} \][/tex]

Substituting the values:

[tex]\[ x_1 = \frac{{-(-9) - \sqrt{53}}}{2 \cdot 7} = \frac{9 - \sqrt{53}}{14} \][/tex]
[tex]\[ x_2 = \frac{{-(-9) + \sqrt{53}}}{2 \cdot 7} = \frac{9 + \sqrt{53}}{14} \][/tex]

3. Round the Solutions to Two Decimal Places:
- [tex]\(x_1\)[/tex] value: [tex]\(\frac{9 - \sqrt{53}}{14} \approx 0.12\)[/tex]
- [tex]\(x_2\)[/tex] value: [tex]\(\frac{9 + \sqrt{53}}{14} \approx 1.16\)[/tex]

Therefore, the solutions to the quadratic equation [tex]\(7x^2 - 9x + 1 = 0\)[/tex] accurate to two decimal places are:

[tex]\[ \text{smaller } x\text{-value} \quad x = 0.12 \][/tex]
[tex]\[ \text{larger } x\text{-value} \quad x = 1.16 \][/tex]